DYMOKE

dymoke, the name of an english family holding the office of king's champion. the functions of the champion were to ride into westminster hall at the coronation banquet, and challenge all comers to impugn the king's title (see champion). the earliest record of the ceremony at the coronation of an english king dates from the accession of richard ii. on this occasion the champion was sir john dymoke (d. 1381), who held the manor of scrivelsby, lincolnshire, in right of his wife margaret, granddaughter of joan ludlow, who was the daughter and co-heiress of philip marmion, last baron marmion. the marmions claimed descent from the lords of fontenay, hereditary champions of the dukes of normandy, and held the castle of tamworth, leicestershire, and the manor of scrivelsby, lincolnshire. the right to the championship was disputed with the dymoke family by sir baldwin de freville, lord of tamworth, who was descended from an elder daughter of philip marmion. the court of claims eventually decided in favour of the owners of scrivelsby on the ground that scrivelsby was held in grand serjeanty, that is, that its tenure was dependent on rendering a special service, in this case the championship. sir thomas dymoke (1428?-1471) joined a lancastrian rising in 1469, and, with his brother-in-law richard, lord willoughby and welles, was beheaded in 1471 by order of edward iv. after he had been induced to leave sanctuary on a promise of personal safety. the estates were restored to his son sir robert dymoke (d. 1546), champion at the coronations of richard iii., henry vii. and henry viii., who distinguished himself at the siege of tournai and became treasurer of the kingdom. his descendants acted as champions at successive coronations. lewis dymoke (d. 1820) put in an unsuccessful claim before the house of lords for the barony of marmion. his nephew henry (1801-1865) was champion at the coronation of george iv. he was accompanied on that occasion by the duke of wellington and lord howard of effingham. henry dymoke was created a baronet; he was succeeded by his brother john, rector of scrivelsby (1804-1873), whose son henry lionel died without issue in 1875, when the baronetcy became extinct, the estate passing to a collateral branch of the family. after the coronation of george iv. the ceremony was allowed to lapse, but at the coronation of king edward vii. h.s. dymoke bore the standard of england in westminster abbey. dynamics (from gr. [greek: dynamis], strength), the name of a branch of the science of mechanics (q.v.). the term was at one time restricted to the treatment of motion as affected by force, being thus opposed to statics, which investigated equilibrium or conditions of rest. in more recent times the word has been applied comprehensively to the action of force on bodies either at rest or in motion, thus including "dynamics" (now termed kinetics) in the restricted sense and "statics." analytical dynamics.--the fundamental principles of dynamics, and their application to special problems, are explained in the articles mechanics and motion, laws of, where brief indications are also given of the more general methods of investigating the properties of a dynamical system, independently of the accidents of its particular constitution, which were inaugurated by j.l. lagrange. these methods, in addition to the unity and breadth which they have introduced into the treatment of pure dynamics, have a peculiar interest in relation to modern physical speculation, which finds itself confronted in various directions with the problem of explaining on dynamical principles the properties of systems whose ultimate mechanism can at present only be vaguely conjectured. in determining the properties of such systems the methods of analytical geometry and of the infinitesimal calculus (or, more generally, of mathematical analysis) are necessarily employed; for this reason the subject has been named analytical dynamics. the following article is devoted to an outline of such portions of general dynamical theory as seem to be most important from the physical point of view. 1. _general equations of impulsive motion._ the systems contemplated by lagrange are composed of discrete particles, or of rigid bodies, in finite number, connected (it may be) in various ways by invariable geometrical relations, the fundamental postulate being that the position of every particle of the system at any time can be completely specified by means of the instantaneous values of a finite number of independent variables q1, q2, ... q_n, each of which admits of continuous variation over a certain range, so that if x, y, z be the cartesian co-ordinates of any one particle, we have for example x = [f](q1, q2, ... q_n), y = &c., z = &c., (1) where the functions [f] differ (of course) from particle to particle. in modern language, the variables q1, q2, ... q_n are _generalized co-ordinates_ serving to specify the _configuration_ of the system; their derivatives with respect to the time are denoted by q`1, q`2, ... q`_n, and are called the _generalized components of velocity_. the continuous sequence of configurations assumed by the system in any actual or imagined motion (subject to the given connexions) is called the _path_. impulsive motion. for the purposes of a connected outline of the whole subject it is convenient to deviate somewhat from the historical order of development, and to begin with the consideration of _impulsive_ motion. whatever the actual motion of the system at any instant, we may conceive it to be generated instantaneously from rest by the application of proper impulses. on this view we have, if x, y, z be the rectangular co-ordinates of any particle m, mx` = x', my` = y', mz` = z', (2) where x', y', z' are the components of the impulse on m. now let [delta]x, [delta]y, [delta]z be any infinitesimal variations of x, y, z which are consistent with the connexions of the system, and let us form the equation [sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = [sigma](x'[delta]x + y'[delta]y + z'[delta]z), (3) where the sign [sigma] indicates (as throughout this article) a summation extending over all the particles of the system. to transform (3) into an equation involving the variations [delta]q1, [delta]q2, ... of the generalized co-ordinates, we have dpx dpx x` = ---- q`1 + ---- q`2 + ..., &c., &c. (4) dpq1 dpq2 dpx dpx [delta]x = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c. (5) dpq1 dpq2 and therefore [sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = (a11q`1 + a12q`2 + ...)[delta]q1 + (a21q`1 + a22q`2 + ...)[delta]q2 + ..., (6) where _ _ | / dpx \ squared / dpy \ squared / dpz \ squared | \ a_rr = [sigma]m | ( ----- ) + ( ----- ) + ( ----- ) |, | |_ \dpq_r/ \dpq_r/ \dpq_r/ _| | _ _ > (7) | dpx dpx dpy dpy dpz dpz | | a_rs = [sigma]m | ----- ----- + ----- ----- + ----- ----- | = a_sr. | |_dpq_r dpq_s dpq_r dpq_s dpq_r dpq_s _| / if we form the expression for the kinetic energy [tau] of the system, we find 2[tau] = [sigma]m(x` squared + y` squared + z` squared) = a11q`1 squared + a22q`2 squared + ... + 2a12q`1q`2 + ... (8) the coefficients a11, a22, ... a12, ... are by an obvious analogy called the _coefficients of inertia_ of the system; they are in general functions of the co-ordinates q1, q2, ... . the equation (6) may now be written dp[tau] dp[tau] [sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = ------- [delta]q1 + ------- [delta]q2 + ... (9) dpq`1 dpq`2 this maybe regarded as the cardinal formula in lagrange's method. for the right-hand side of (3) we may write [sigma](x'[delta]x + y'[delta]y + z'[delta]z) = q'1[delta]q1 + q'2[delta]q2 + ... , (10) where / dpx dpy dpz \ q'_r = [sigma]( x'----- + y'----- + z'----- ). (11) \ dpq_r dpq_r dpq_r/ the quantities q1, q2, ... are called the _generalized components of impulse_. comparing (9) and (10), we have, since the variations [delta]q1, [delta]q2,... are independent, dp[tau] dp[tau] ------- = q'1, ------- = q'2, ... (12) dpq`1 dpq`2 these are the general equations of impulsive motion. it is now usual to write dp[tau] p_r = ------- (13) dpq`_r the quantities p1, p2, ... represent the effects of the several component impulses on the system, and are therefore called the _generalized components of momentum_. in terms of them we have [sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = p1[delta]q1 + p2[delta]q2 + ... (14) also, since [tau] is a homogeneous quadratic function of the velocities q`1, q`2 ..., 2[tau] = p1q`1 + p2q`2 + ... (15) this follows independently from (14), assuming the special variations [delta]x = x`dt, &c., and therefore [delta]q1 = q`1dt, [delta]q2 = q`2dt, ... reciprocal theorems. again, if the values of the velocities and the momenta in any other motion of the system through the same configuration be distinguished by accents, we have the identity p1q`'1 + p2q`'2 + ... = p'1q`1 + p'2q`2 + ..., (16) each side being equal to the symmetrical expression a11q`1q''1 + a22q`2q`'2 + ... + a12(q`1q`'2 + q`'1q`2) + ... (17) the theorem (16) leads to some important reciprocal relations. thus, let us suppose that the momenta p1, p2, ... all vanish with the exception of p1, and similarly that the momenta p'1, p'2, ... all vanish except p'2. we have then p1q`'1 = p'2q`2, or q`2 : p1 = q`'1 : p'2 (18) the interpretation is simplest when the co-ordinates q1, q2 are both of the same kind, e.g. both lines or both angles. we may then conveniently put p1 = p'2, and assert that the velocity of the first type due to an impulse of the second type is equal to the velocity of the second type due to an equal impulse of the first type. as an example, suppose we have a chain of straight links hinged each to the next, extended in a straight line, and free to move. a blow at right angles to the chain, at any point p, will produce a certain velocity at any other point q; the theorem asserts that an equal velocity will be produced at p by an equal blow at q. again, an impulsive couple acting on any link a will produce a certain angular velocity in any other link b; an equal couple applied to b will produce an equal angular velocity in a. also if an impulse f applied at p produce an angular velocity [omega] in a link a, a couple fa applied to a will produce a linear velocity [omega]a at p. historically, we may note that reciprocal relations in dynamics were first recognized by h.l.f. helmholtz in the domain of acoustics; their use has been greatly extended by lord rayleigh. velocities in terms of momenta. the equations (13) determine the momenta p1, p2,... as linear functions of the velocities q`1, q`2,... solving these, we can express q`1, q`2 ... as linear functions of p1, p2,... the resulting equations give us the velocities produced by any given system of impulses. further, by substitution in (8), we can express the kinetic energy as a homogeneous quadratic function of the momenta p1, p2,... the kinetic energy, _as so expressed_, will be denoted by [tau]'; thus 2[tau]' = a'11p1 squared + a'22p2 squared + ... + 2a'12p - p2 + ... (19) where a'11, a'22,... a'12,... are certain coefficients depending on the configuration. they have been called by maxwell the _coefficients of mobility_ of the system. when the form (19) is given, the values of the velocities in terms of the momenta can be expressed in a remarkable form due to sir w.r. hamilton. the formula (15) may be written p1q`1 + p2q`2 + ... = [tau] + [tau]', ... (20) where [tau] is supposed expressed as in (8), and [tau]' as in (19). hence if, for the moment, we denote by [delta] a variation affecting the velocities, and therefore the momenta, but not the configuration, we have p1[delta]q`1 + q`1[delta]p + p2[delta]q`2 + q`2[delta]p2 + ... = [delta][tau] + [delta][tau]' dp[tau] dp[tau] dp[tau]' dp[tau]' = ------- [delta]q`1 + ------- [delta]q`2 + ... + -------- [delta]p1 + -------- [delta]p2 + ... (21) dpq`1 dpq`2 dpp1 dpp2 in virtue of (13) this reduces to dp[tau]' dp[tau]' q`1[delta]p1 + q`2[delta]p2 + ... = ------- [delta]p1 + ------- [delta]p2 + ... (22) dpp1 dpp2 since [delta]p1, [delta]p2, ... may be taken to be independent, we infer that dp[tau]' dp[tau]' q`1 = -------, q`2 = -------, ... (23) dpp1 dpp2 in the very remarkable exposition of the matter given by james clerk maxwell in his _electricity and magnetism_, the hamiltonian expressions (23) for the velocities in terms of the impulses are obtained directly from first principles, and the formulae (13) are then deduced by an inversion of the above argument. routh's modification. an important modification of the above process was introduced by e.j. routh and lord kelvin and p.g. tait. instead of expressing the kinetic energy in terms of the velocities alone, or in terms of the momenta alone, we may express it in terms of the velocities corresponding to some of the co-ordinates, say q1, q2, ... q_m, and of the momenta corresponding to the remaining co-ordinates, which (for the sake of distinction) we may denote by [chi], [chi]', [chi]", .... thus, [tau] being expressed as a homogeneous quadratic function of q`1, q`2, ... q`_m, [chi]`, [chi]`', [chi]`", ..., the momenta corresponding to the co-ordinates [chi], [chi]', [chi]", ... may be written dp[tau] dp[tau] dp[tau] [kappa] = --------, [kappa]' = ---------, [kappa]" = ------------, ... (24) dp[chi]` dp[chi]`' dp[.[chi]`" these equations, when written out in full, determine [chi]`, [chi]`', [chi]`", ... as linear functions of q`1, q`2, ... q`_m, [kappa], [kappa]', [kappa]",... we now consider the function r = [tau] - [kappa][chi]' - [kappa]'[chi]]`' - [kappa]"[chi]]`" - ..., (25) supposed expressed, by means of the above relations in terms of q`1, q`2, ... q`_m, [kappa], [kappa]', [kappa]",... performing the operation [delta] on both sides of (25), we have dpr dpr dp[tau] dp[tau] ----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... + -------- [delta][chi]` + ... dpq`1 dp[kappa] dpq`1 dp[chi]` - [kappa]dp[chi]` - [chi]`[delta][kappa] - ... , (26) where, for brevity, only one term of each type has been exhibited. omitting the terms which cancel in virtue of (24), we have dpr dpr dp[tau] ----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... - [chi]`[delta][kappa] - ... (27) dpq`1 dp[kappa] dpq`1 since the variations [delta]q1, [delta]q2, ... [delta]q_m, [delta][kappa], [delta][kappa]', [delta][kappa]", ... may be taken to be independent, we have dp[tau] dpr dp[tau] dpr p1 = ------- = -----, p2 = ------- = -----, ... (28) dpq`1 dpq`1 dpq`2 dpq`2 and dpr dpr dpr [chi]` = - ---------, [chi]`' = - ----------, [chi]]`" = - ---------, ... (29) dp[kappa] dp[kappa]' dp[kappa]" an important property of the present transformation is that, when expressed in terms of the new variables, the kinetic energy is the sum of two homogeneous quadratic functions, thus [tau] = [@] + k, (30) where [@] involves the velocities q`1, q`2, ... q`_m alone, and k the momenta [kappa], [kappa]', [kappa]", ... alone. for in virtue of (29) we have, from (25), / dpr dpr dpr \ [tau] = r - ( [kappa] --------- + [kappa]' ---------- + [kappa]" ----------- + ... ), (31) \ dp[kappa] dp[kappa]' dp[kappa]" / and it is evident that the terms in r which are bilinear in respect of the two sets of variables q`1, q`2, ... q`_m and [kappa], [kappa]', [kappa]", ... will disappear from the right-hand side. maximum and minimum energy. it may be noted that the formula (30) gives immediate proof of two important theorems due to bertrand and to lord kelvin respectively. let us suppose, in the first place, that the system is started by given impulses of certain types, but is otherwise free. j.l.f. bertrand's theorem is to the effect that the kinetic energy is _greater_ than if by impulses of the remaining types the system were constrained to take any other course. we may suppose the co-ordinates to be so chosen that the constraint is expressed by the vanishing of the velocities q`1, q`2, ... q`_m, whilst the given impulses are [kappa], [kappa]', [kappa]",... hence the energy in the actual motion is greater than in the constrained motion by the amount [@]. again, suppose that the system is started with prescribed velocity components q`1, q`2, ... q`_m, by means of proper impulses of the corresponding types, but is otherwise free, so that in the motion actually generated we have [kappa] = 0, [kappa]' = 0, [kappa]" = 0, ... and therefore k = 0. the kinetic energy is therefore _less_ than in any other motion consistent with the prescribed velocity-conditions by the value which k assumes when [kappa], [kappa]', [kappa]", ... represent the impulses due to the constraints. simple illustrations of these theorems are afforded by the chain of straight links already employed. thus if a point of the chain be held fixed, or if one or more of the joints be made rigid, the energy generated by any given impulses is less than if the chain had possessed its former freedom. 2. _continuous motion of a system._ lagrange's equations. we may proceed to the continuous motion of a system. the equations of motion of any particle of the system are of the form mx" = x, my" = y, mz" = z (1) now let x + [delta]x, y + [delta]y, z + [delta]z be the co-ordinates of m in any arbitrary motion of the system differing infinitely little from the actual motion, and let us form the equation [sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = [sigma](x[delta]x + y[delta]y + z[delta]z) (2) lagrange's investigation consists in the transformation of (2) into an equation involving the independent variations [delta]q1, [delta]q2, ... [delta]q_n. it is important to notice that the symbols [delta] and d/dt are commutative, since d dx d [delta]x` = --(x + [delta]x) - -- = --[delta]x, &c. (3) dt dt dt hence d [sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = -- [sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) dt - [sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`) d = --(p1[delta]q1 + p2[delta]q2 + ...) - [delta][tau], (4) dt by sec. 1 (14). the last member may be written p`1[delta]q1 + p1[delta]q`1 + p`2[delta]q2 + p2[delta]q`2 + ... dp[tau] dp[tau] dp[tau] dp[tau] - ------- [delta]q`1 - ------- [delta]q1 - ------- [delta]q`2 - ------- [delta]q2 - ... (5) dpq`1 dpq1 dpq`2 dpq2 hence, omitting the terms which cancel in virtue of sec. 1 (13), we find / dp[tau]\ / dp[tau]\ [sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = (p`1 - ------- ) [delta]q1 + (p`2 - ------- ) [delta]q2 + ... (6) \ dpq1 / \ dpq2 / for the right-hand side of (2) we have [sigma](x[delta]x + y[delta]y + z[delta]z) = q1[delta]q1 + q2[delta]q2 + ..., (7) / dpx dpy dpz \ where q_r = [sigma]( x ----- + y ----- + z ----- ) (8) \ dpq_r dpq_r dpq_r/ the quantities q1, q2, ... are called the _generalized components of force_ acting on the system. comparing (6) and (7) we find dp[tau] dp[tau] p`1 - ------- = q1, p`2 - ------- = q2, ..., (9) dpq`1 dpq`2 or, restoring the values of p1, p2, ..., d /dp[tau]\ dp[tau] d /dp[tau]\ dp[tau] -- ( ------- ) - ------- = q1, -- ( ------- ) - ------- = q2, ... (10) dt \ dpq`1 / dpq1 dt \ dpq`2 / dpq2 these are lagrange's general equations of motion. their number is of course equal to that of the co-ordinates q1, q2, ... to be determined. analytically, the above proof is that given by lagrange, but the terminology employed is of much more recent date, having been first introduced by lord kelvin and p.g. tait; it has greatly promoted the physical application of the subject. another proof of the equations (10), by direct transformation of co-ordinates, has been given by hamilton and independently by other writers (see mechanics), but the variational method of lagrange is that which stands in closest relation to the subsequent developments of the subject. the chapter of maxwell, already referred to, is a most instructive commentary on the subject from the physical point of view, although the proof there attempted of the equations (10) is fallacious. in a "conservative system" the work which would have to be done by extraneous forces to bring the system from rest in some standard configuration to rest in the configuration (q1, q2, ... q_n) is independent of the path, and may therefore be regarded as a definite function of q1, q2, ... q_n. denoting this function (the _potential energy_) by v, we have, if there be no extraneous force on the system, [sigma](x[delta]x + y[delta]y + z[delta]z) = - [delta]v, (11) and therefore dpv dpv q1 = - ----, q2 = - ----, .... (12) dpq1 dpq2 hence the typical lagrange's equation may be now written in the form d /dp[tau]\ dp[tau] dpv -- ( ------- ) - ------- = - -----, (13) dt \dpq`_r / dpq_r dpq_r or, again, dp p`_r = - ----- (v - [tau]) (14) dpq_r it has been proposed by helmholtz to give the name _kinetic potential_ to the combination v - [tau]. as shown under mechanics, sec. 22, we derive from (10) d[tau] ------ = q1q`1 + q2q`2 + ..., (15) dt and therefore in the case of a conservative system free from extraneous force, d --([tau] + v) = 0 or [tau] + v = const., (16) dt which is the equation of energy. for examples of the application of the formula (13) see mechanics, sec. 22. 3. _constrained systems._ case of varying relations. it has so far been assumed that the geometrical relations, if any, which exist between the various parts of the system are of the type sec. 1 (1), and so do not contain t explicitly. the extension of lagrange's equations to the case of "varying relations" of the type x = f(t, q1, q2,...q_n), y = &c., z = &c., (1) was made by j.m.l. vieille. we now have dpx dpx dpx x` = --- + ---- q`1 + ---- q`2 + ..., &c., &c., (2) dpt dpq1 dpq2 dpx dpx dpx = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c., (3) dpq1 dpq2 so that the expression sec. 1 (8) for the kinetic energy is to be replaced by 2[tau] = [alpha]0 + 2[alpha]1q`1 + 2[alpha]2q`2 + ... + a11q`1 squared + a22q`2 squared + ... + a12q`1q`2 + ..., (4) where _ _ \ | /dpx\ squared /dpy\ squared /dpz\ squared | | a0 = [sigma]m |( --- ) + ( --- ) + ( --- ) |, | |_\dpt/ \dpt/ \dpt/ _| | _ _ > (5) | dpx dpx dpy dpy dpz dpz | | a_r = [sigma]m | --- ----- + --- ----- + --- ----- |, | |_dpt dpq_r dpt dpq_r dpt dpq_r_| | / and the forms of a_rr, a_rs are as given by sec. 1 (7). it is to be remembered that the coefficients [alpha]0, [alpha]1, [alpha]2, ... a11, a22, ... a12 ... will in general involve t explicitly as well as implicitly through the co-ordinates q1, q2,... again, we find [sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = ([alpha]1 + a11q`1 + a12q`2 + ...)[delta]q1 + ([alpha]2 + a21q`1 + a22q`2 + ...)dpq2 + ... dp[tau] dp[tau] = ------- [delta]q1 + ------- [delta]q2 + ... dpq`1 dpq`2 = p1[delta]q1 + p2[delta]q2 + ..., (6) where p_r is defined as in sec. 1 (13). the derivation of lagrange's equations then follows exactly as before. it is to be noted that the equation sec. 2 (15) does not as a rule now hold. the proof involved the assumption that [tau] is a homogeneous quadratic function of the velocities q`1, q`2.... it has been pointed out by r.b. hayward that vieille's case can be brought under lagrange's by introducing a new co-ordinate ([chi]) in place of t, so far as it appears explicitly in the relations (1). we have then 2[tau] = [alpha]0[chi]` squared + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]` + a11q`1 squared + a22q`2 squared + ... + 2a12q`1q`2 + .... (7) the equations of motion will be as in sec. 2 (10), with the additional equation d dp[tau] dp[tau] -- -------- - ------- = x, (8) dt dp[chi]` dp[chi] where x is the force corresponding to the co-ordinate [chi]. we may suppose x to be adjusted so as to make [chi]" = 0, and in the remaining equations nothing is altered if we write t for [chi] before, instead of after, the differentiations. the reason why the equation sec. 2 (15) no longer holds is that we should require to add a term x[chi]` on the right-hand side; this represents the rate at which work is being done by the constraining forces required to keep [chi]` constant. as an example, let x, y, z be the co-ordinates of a particle relative to axes fixed in a solid which is free to rotate about the axis of z. if [phi] be the angular co-ordinate of the solid, we find without difficulty 2[tau] = m(x` squared + y` squared +z` squared) + 2[phi]`m(xy` - yx`) + {i + m(x squared + y squared)}[phi]` squared, (9) where i is the moment of inertia of the solid. the equations of motion, viz. d dp[tau] dp[tau] d dp[tau] dp[tau] d dp[tau] dp[tau] -- ------ - ------ = x, -- ------- - ------- = y, -- ------- - ------- = z, (10) dt dpx` dpx dt dpy` dpy dt dpz` dpz d dp[tau] dp[tau] and -- -------- - ------- = [phi], (11) dt dp[phi]` dp[phi] become m(x" - 2[phi]`y` - x[phi]` squared - y[phi]") = x, m(y" + 2[phi]`x` - y[phi]` squared + x[phi]`) = y, mz" = z, (12) _ _ d | / \ | and -- |(i + m(x squared + y squared)) [phi]` + m(xy` - yx`)| = [phi]. (13) dt |_\ / _| if we suppose [phi] adjusted so as to maintain [phi]" = 0, or (again) if we suppose the moment of inertia i to be infinitely great, we obtain the familiar equations of motion relative to moving axes, viz. m(x" - 2[omega]y` - [omega] squaredx) = x, m(y" + 2[omega]x` - [omega] squaredy) = y, mz" = z, (14) where [omega] has been written for [phi]. these are the equations which we should have obtained by applying lagrange's rule at once to the formula 2[tau] = m(x` squared + y` squared + z` squared) + 2m[omega](xy` - yx`) + m[omega] squared(x squared + y squared), (15) which gives the kinetic energy of the particle referred to axes rotating with the constant angular velocity [omega]. (see mechanics, sec. 13.) more generally, let us suppose that we have a certain group of co-ordinates [chi], [chi]', [chi]", ... whose absolute values do not affect the expression for the kinetic energy, and that by suitable forces of the corresponding types the velocity-components [chi]`, [chi]`', [chi]`", ... are maintained constant. the remaining co-ordinates being denoted by q1, q2, ... q_n, we may write 2[tau] = [@] + [tau]0 + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]` + 2([alpha]'1q`1 + [alpha]'2q`2 + ...)[chi]`' + ..., (16) where [@] is a homogeneous quadratic function of the velocities q`1, q`2, ... q`_n of the type sec.1 (8), whilst [tau]0 is a homogeneous quadratic function of the velocities [chi]`,[chi]`', [chi]`", ... alone. the remaining terms, which are bilinear in respect of the two sets of velocities, are indicated more fully. the formulae (10) of sec. 2 give n equations of the type d /dp[@]\ /dp[@]\ dp[tau]0 --( ----- ) - ( ----- ) + (r, 1)q`1 + (r, 2)q`2 + ... - -------- = q_r (17) dt \dpq_r/ \dpq_r/ dpq_r where /dpa_r dpa_s\ /dpa'_r dpa'_s\ (r, s) = ( ----- - ----- )[chi]` + ( ------ - ------ )[chi]`' + .... (18) \dpq_s dpq_r/ \dpq_s dpq_r/ these quantities (r, s) are subject to the relations (r, s) = -(s, r), (r, r) = 0 (19) the remaining dynamical equations, equal in number to the co-ordinates [chi], [chi]', [chi]", ..., yield expressions for the forces which must be applied in order to maintain the velocities [chi]`, [chi]`', [chi]`", ... constant; they need not be written down. if we follow the method by which the equation of energy was established in sec. 2, the equations (17) lead, on taking account of the relations (19), to d --([@] - [tau]0) = q1q`1 + q2q`2 + ... + q_nq`_n, (20) dt or, in case the forces q_r depend only on the co-ordinates q1, q2, ... q_n and are conservative, [@] + v - [tau]0 = const. (21) the conditions that the equations (17) should be satisfied by zero values of the velocities q`1, q`2, ... q`_n are dp[tau]0 q_r = - --------, (22) dpq_r or in the case of conservative forces dp ------ (v - [tau]0) = 0, (23) dpq_r i.e. the value of v - [tau]0 must be _stationary_. rotating axes. we may apply this to the case of a system whose configuration relative to axes rotating with constant angular velocity ([omega]) is defined by means of the n co-ordinates q1, q2, ... q_n. this is important on account of its bearing on the kinetic theory of the tides. since the cartesian co-ordinates x, y, z of any particle m of the system relative to the moving axes are functions of q1, q2, ... q_n, of the form sec. 1 (1), we have, by (15) 2[@] = [sigma]m(x` squared + y` squared + z` squared), 2[tau]0 = [omega] squared[sigma]m(x squared + y squared), (24) / dpy dpx \ a_r= [sigma]m( x----- - y----- ), (25) \ dpq_r dpq_r/ whence dp(x, y) (r, s) = 2[omega].[sigma]m ------------. (26) dp(q_s, q_r) the conditions of relative equilibrium are given by (23). it will be noticed that this expression v - [tau]0, which is to be stationary, differs from the true potential energy by a term which represents the potential energy of the system in relation to fictitious "centrifugal forces." the question of stability of relative equilibrium will be noticed later (sec. 6). it should be observed that the remarkable formula (20) may in the present case be obtained directly as follows. from (15) and (14) we find d[tau] d ------ = --([@] + [tau]0) + [omega].[sigma]m(xy" - yx") dt dt d = --([@] - [tau]0) + [omega].[sigma](xy - yx). (27) dt this must be equal to the rate at which the forces acting on the system do work, viz. to [omega][sigma](xy - yx) + q1q`1 + q2q`2 + ... + q_nq`_n, where the first term represents the work done in virtue of the rotation. constrained systems. we have still to notice the modifications which lagrange's equations undergo when the co-ordinates q1, q2, ... q_n are not all independently variable. in the first place, we may suppose them connected by a number m ( < n) of relations of the type a(t, q1, q2, ... q_n) = 0, b(t, q1, q2, ... q_n) = 0, &c. (28) these may be interpreted as introducing partial constraints into a previously free system. the variations [delta]q1, [delta]q2, ... [delta]q_n in the expressions (6) and (7) of sec. 2 which are to be equated are no longer independent, but are subject to the relations dpa dpa dpb dpb ---- [delta]q1 + ---- [delta]q2 + ... = 0, ---- [delta]q1 + ---- [delta]q2 + ... = 0, &c. (29) dpq1 dpq2 dpq1 dpq2 introducing indeterminate multipliers [lambda], mu, ..., one for each of these equations, we obtain in the usual manner n equations of the type d dp[tau] dp[tau] dpa dpb -- ------- - ------- = q_r + [lambda] ----- + mu ----- + ..., (30) dt dpq`_r dpq_r dpq_r dpq_r in place of sec. 2 (10). these equations, together with (28), serve to determine the n co-ordinates q1, q2, ... q_n and the m multipliers [lambda], mu, .... when t does not occur explicitly in the relations (28) the system is said to be _holonomic_. the term connotes the existence of integral (as opposed to differential) relations between the co-ordinates, independent of the time. again, it may happen that although there are no prescribed relations between the co-ordinates q1, q2, ... q_n, yet from the circumstances of the problem certain geometrical conditions are imposed on their _variations_, thus a1[delta]q1 + a2[delta]q2 + ... = 0, b1[delta]q1 + b2[delta]q2 + ... = 0, &c., (31) where the coefficients are functions of q1, q2, ... q_n and (possibly) of t. it is assumed that these equations are not integrable as regards the variables q1, q2, ... q_n; otherwise, we fall back on the previous conditions. cases of the present type arise, for instance, in ordinary dynamics when we have a solid rolling on a (fixed or moving) surface. the six co-ordinates which serve to specify the position of the solid at any instant are not subject to any necessary relation, but the conditions to be satisfied at the point of contact impose three conditions of the form (31). the general equations of motion are obtained, as before, by the method of indeterminate multipliers, thus d dp[tau] dp[tau] -- ------- - ------- = q_r + [lambda]a_r + mub_r + ... (32) dt dpq`_r dpq_r the co-ordinates q1, q2, ... q_n, and the indeterminate multipliers [lambda], mu, ..., are determined by these equations and by the velocity-conditions corresponding to (31). when t does not appear explicitly in the coefficients, these velocity-conditions take the forms a1q`1 + a2q`2 + ... = 0, b1q`1 + b2q`2 + ... = 0, &c. (33) systems of this kind, where the relations (31) are not integrable, are called _non-holonomic_. 4. _hamiltonian equations of motion._ in the hamiltonian form of the equations of motion of a conservative system with unvarying relations, the kinetic energy is supposed expressed in terms of the _momenta_ p1, p2, ... and the co-ordinates q1, q2, ..., as in sec. 1 (19). since the symbol [delta] now denotes a variation extending to the co-ordinates as well as to the momenta, we must add to the last member of sec. 1 (21) terms of the types dp[tau] dp[tau]' ------- [delta]q1 + -------- [delta]q1 + ... (1) dpq1 dpq1 since the variations [delta]p1, [delta]p2, ... [delta]q1, [delta]q2, ... may be taken to be independent, we infer the equations sec. 1 (23) as before, together with dp[tau] dp[tau]' dp[tau] dp[tau]' ------ = - --------, ------- = - --------, ..., (2) dpq1 dpq1 dpq2 dpq2 hence the lagrangian equations sec. 2 (14) transform into dp dp p`1 = - ----([tau]' + v), p`2 = ---- ([tau]' + v), ... (3) dpq1 dpq2 if we write h = [tau]' + v, (4) so that h denotes the _total energy_ of the system, supposed expressed in terms of the new variables, we get dph dph p`1 = - ----, p`2 = - ----, ... (5) dpq1 dpq2 if to these we join the equations dph dph q`1 = ----, q`2 = ----, ..., (6) dpp1 dpp2 which follow at once from sec. 1 (23), since v does not involve p1, p2, ..., we obtain a complete system of differential equations _of the first order_ for the determination of the motion. the equation of energy is verified immediately by (5) and (6), since these make dh dph dph dph dph -- = ---- p`1 + ---- p`2 + ... + ---- q`1 + ---- q`2 + ... = 0. (7) dt dpp1 dpp2 dpq1 dpq2 the hamiltonian transformation is extended to the case of varying relations as follows. instead of (4) we write h = p1q`1 + p2q`2 + ... - [tau] + v, (8) and imagine h to be expressed in terms of the momenta p1, p2, ..., the co-ordinates q1, q2, ..., and the time. the internal forces of the system are assumed to be conservative, with the potential energy v. performing the variation [delta] on both sides, we find dp[tau] dpv [delta]h = q`1[delta]p1 + ... - ------- [delta]q1 + ---- [delta]q + ..., (9) dpq1 dpq1 terms which cancel in virtue of the definition of p1, p2, ... being omitted. since [delta]p1, [delta]p2, ..., [delta]q1, [delta]q2, ... may be taken to be independent, we infer dph dph q`1 = ----, q`2 = ----, ..., (10) dpp1 dpp2 and dp dph dp dph ---- ([tau] - v) = - ----, ----([tau] - v) = - ----, .... (11) dpq1 dpq1 dpq2 dpq2 it follows from (11) that dph dph p`1 = - ----, p`2 = - ----, .... (12) dpq1 dpq2 the equations (10) and (12) have the same form as above, but h is no longer equal to the energy of the system. 5. _cyclic systems._ a _cyclic_ or _gyrostatic_ system is characterized by the following properties. in the first place, the kinetic energy is not affected if we alter the absolute values of certain of the co-ordinates, which we will denote by [chi], [chi]', [chi]", ..., provided the remaining co-ordinates q1, q2, ... q_m and the velocities, including of course the velocities [.[chi]], [.[chi]]', [.[chi]]", ..., are unaltered. secondly, there are no forces acting on the system of the types [chi], [chi]', [chi]", .... this case arises, for example, when the system includes gyrostats which are free to rotate about their axes, the co-ordinates [chi], [chi]', [chi]", ... then being the angular co-ordinates of the gyrostats relatively to their frames. again, in theoretical hydrodynamics we have the problem of moving solids in a frictionless liquid; the ignored co-ordinates [chi], [chi]', [chi]", ... then refer to the fluid, and are infinite in number. the same question presents itself in various physical speculations where certain phenomena are ascribed to the existence of _latent motions_ in the ultimate constituents of matter. the general theory of such systems has been treated by e.j. routh, lord kelvin, and h.l.f. helmholtz. routh's equations. if we suppose the kinetic energy [tau] to be expressed, as in lagrange's method, in terms of the co-ordinates and the velocities, the equations of motion corresponding to [chi], [chi]', [chi]'', ... reduce, in virtue of the above hypotheses, to the forms d dp[tau] d dp[tau] d dp[tau] -- --------- = 0, -- --------- = 0, -- --------- = 0, ..., (1) dt dp[chi]` dt dp[chi]`' dt dp[chi]`" whence dp[tau] dp[tau] dp[tau] -------- = [kappa], --------- = [kappa]', --------- = [kappa]", ..., (2) dp[chi]` dp[chi]`' dp[chi]`" where [kappa], [kappa]', [kappa]", ... are the constant momenta corresponding to the cyclic co-ordinates [chi], [chi]', [chi]", .... these equations are linear in [.[chi]], [.[chi]]', [.[chi]]", ...; solving them with respect to these quantities and substituting in the remaining lagrangian equations, we obtain m differential equations to determine the remaining co-ordinates q1, q2, ... q_m. the object of the present investigation is to ascertain the general form of the resulting equations. the retained co-ordinates q1, q2, ... q_m may be called (for distinction) the _palpable_ co-ordinates of the system; in many practical questions they are the only co-ordinates directly in evidence. if, as in sec. 1 (25), we write r = [tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ..., (3) and imagine r to be expressed by means of (2) as a quadratic function of q`1, q`2, ... q`_m, [kappa], [kappa]', [kappa]", ... with coefficients which are in general functions of the co-ordinates q1, q2, ... q_m, then, performing the operation [delta] on both sides, we find dpr dpr dpr dp[tau] dp[tau] -----[delta]q`1 + ... + ---------[delta][kappa] + ... + ----[delta]q1 + ... = -------[delta]q`1 + ... + -------[delta]q1 + ... dpq`1 dp[kappa] dpq1 dpq`1 dpq1 dp[tau] dp[tau] + --------[delta][chi]` + ... + --------[delta]q1 + ... - [kappa][delta][chi]` - [chi]`[delta][kappa] - .... (4) dp[chi]` dp[chi]1 omitting the terms which cancel by (2), we find dp[tau] dpr dp[tau] dpr ------- = -----, ------- = -----, ..., (5) dpq`1 dpq`1 dpq`2 dpq`2 dp[tau] dpr dp[tau] dpr ------- = ----, ------- = ----, ..., (6) dpq1 dpq1 dpq2 dpq2 dpr dpr dpr [chi]` = - ---------, [chi]`' = - ----------, [chi]`" = - ----------, ... (7) dp[kappa] dp[kappa]' dp[kappa]" substituting in sec. 2 (10), we have d dpr dpr d dpr dpr -- ----- - ----- = q1, -- ----- - ---- = q2, ... (8) dt dpq`1 dpq1 dt dpq`2 dpq2 these are routh's forms of the modified lagrangian equations. equivalent forms were obtained independently by helmholtz at a later date. kelvin's equations. the function r is made up of three parts, thus r = r(2,0) + r(1,1) + r(0,2), ... (9) where r(2,0) is a homogeneous quadratic function of q`1, q`2, ... q`_m, r(0,2) is a homogeneous quadratic function of [kappa], [kappa]', [kappa]", ..., whilst r(1,1) consists of products of the velocities q`1, q`2, ... q`_m into the momenta [kappa], [kappa]', [kappa]".... hence from (3) and (7) we have / dpr dpr dpr \ [tau] = r - ( [kappa] --------- + [kappa]'---------- + [kappa]" ---------- + ...) \ dp[kappa] dp[kappa]' dp[kappa]" / = r(2,0) - r(0,2). (10) if, as in sec. 1 (30), we write this in the form [tau] = [@] + [kappa], (11) then (3) may be written r = [@] - [kappa] + ss1q`1 + ss2q`2 + ..., (12) where ss1, ss2, ... are linear functions of [kappa], [kappa]', [kappa]", ..., say ss_r = [alpha]_r[kappa] + [alpha]'_r[kappa]' + [alpha]"_r[kappa]" + ..., (13) the coefficients [alpha]_r, [alpha]'_r, [alpha]"_r, ... being in general functions of the co-ordinates q1, q2, ... q_m. evidently ss_r denotes that part of the momentum-component dpr/dpq`_r which is due to the cyclic motions. now d dpr d / dp[@] \ d dp[@] dpss_r dpss_r -- ------ = -- ( ------ + ss_r) = -- ------ + -----q`1 + -----q`2 + ..., (14) dt dpq`_r dt \dpq`_r / dt dpq`_r dpq1 dpq2 dpr dp[@] dp[kappa] dpss1 dpss2 ----- = ----- - --------- + -----q`1 + -----q`2 + .... (15) dpq_r dpq_r dpq_r dpq_r dpq_r hence, substituting in (8), we obtain the typical equation of motion of a gyrostatic system in the form d dp[@] dp[@] dp[kappa] -- ------ - ----- + (r, 1)q`1 + (r, 2)q`2 + ... + (r, s)q`_s + ... + --------- = q_r, (16) dt dpq`_r dpq_r dpq_r where dpss_r dpss_s (r, s) = ----- - -----. (17) dpq_s dpq_r this form is due to lord kelvin. when q1, q2, ... q_m have been determined, as functions of the time, the velocities corresponding to the cyclic co-ordinates can be found, if required, from the relations (7), which may be written dp[kappa] \ [chi]` = --------- - [alpha]1q`1 - [alpha]2q`2 - ..., | dp[kappa] | | dp[kappa] > (18) [chi]`' = ---------- - [alpha]'1q`1 - [alpha]'2q`2 - ..., | dp[kappa]' | | &c., &c. / it is to be particularly noticed that (r, r) = 0, (r, s) = -(s, r). (19) hence, if in (16) we put r = 1, 2, 3, ... m, and multiply by q`1, q`2, ... q`_m respectively, and add, we find d --([@] + [kappa]) = q1q`1 + q2q`2 + ..., (20) dt or, in the case of a conservative system [@] + v + [kappa] = const., (21) which is the equation of energy. the equation (16) includes sec. 3 (17) as a particular case, the eliminated co-ordinate being the angular co-ordinate of a rotating solid having an infinite moment of inertia. in the particular case where the cyclic momenta [kappa], [kappa]', [kappa]", ... are all zero, (16) reduces to d dp[@] dp[@] -- ------ - ----- = q_r. (22) dt dpq`_r dpq_r the form is the same as in sec. 2, and the system now behaves, as regards the co-ordinates q1, q2, ... q_m, exactly like the acyclic type there contemplated. these co-ordinates do not, however, now fix the position of every particle of the system. for example, if by suitable forces the system be brought back to its initial configuration (so far as this is defined by q1, q2, ..., q_m), after performing any evolutions, the ignored co-ordinates [chi], [chi]', [chi]", ... will not in general return to their original values. if in lagrange's equations sec. 2 (10) we reverse the sign of the time-element dt, the equations are unaltered. the motion is therefore reversible; that is to say, if as the system is passing through any configuration its velocities q`1, q`2, ..., q`_m be all reversed, it will (if the forces be the same in the same configuration) retrace its former path. but it is important to observe that the statement does not in general hold of a gyrostatic system; the terms of (16), which are linear in q`1, q`2, ..., q`_m, change sign with dt, whilst the others do not. hence the motion of a gyrostatic system is not reversible, unless indeed we reverse the cyclic motions as well as the velocities q`1, q`2, ..., q`_m. for instance, the precessional motion of a top cannot be reversed unless we reverse the spin. kinetostatics. the _conditions of equilibrium_ of a system with latent cyclic motions are obtained by putting q`1 = 0, q`2 = 0, ... q`_m = 0 in (16); viz. they are dp[kappa] dp[kappa] q1 = ---------, q2 = ---------, ... (23) dpq1 dpq2 these may of course be obtained independently. thus if the system be guided from (apparent) rest in the configuration (q1, q2, ... q_m) to rest in the configuration q1 + [delta]q1, q2 + [delta]q2, ..., q_m + [delta]q_m, the work done by the forces must be equal to the increment of the kinetic energy. hence q1[delta]q1 + q2[delta]q2 + ... = [delta][kappa], (24) which is equivalent to (23). the conditions are the same as for the equilibrium of a system without latent motion, but endowed with potential energy [kappa]. this is important from a physical point of view, as showing how energy which is apparently potential may in its ultimate essence be kinetic. by means of the formulae (18), which now reduce to dp[kappa] dp[kappa] dp[kappa] [chi]` = ---------, [chi]`' = ----------, [chi]`" = ---------- ..., (25) dp[kappa] dp[kappa]' dp[kappa]" [kappa] may also be expressed as a homogeneous quadratic function of the cyclic velocities [.[chi]], [.[chi]]', [.[chi]]", ... denoting it in this form by [tau]0, we have [delta]([tau]0 + [kappa] = 2[delta][kappa] = [delta]([kappa] [chi]` + [kappa]'[chi]`' + [kappa]"[chi]`" + ...). (26) performing the variations, and omitting the terms which cancel by (2) and (25), we find dp[tau]0 dp[kappa] dp[tau]0 dp[kappa] -------- = - ---------, -------- = - ---------, ..., (27) dpq1 dpq1 dpq2 dpq2 so that the formulae (23) become dp[tau]0 dp[tau]0 q1 = - --------, q2 = - --------, ... (28) dpq1 dpq2 a simple example is furnished by the top (mechanics, sec. 22). the cyclic co-ordinates being [psi], [phi], we find ( mu - [nu] cos [theta]) squared [nu] squared 2[@] = a[theta]` squared, 2[kappa] = ----------------------- + -----, a sin squared [theta] c 2[tau]0 = a sin squared[theta][psi]` squared + c([phi]` + [psi] cos [theta]) squared, (29) whence we may verify that dp[tau]0/dp[theta] = - dp[kappa]/dp[theta] in accordance with (27). and the condition of equilibrium dp[kappa] dpv --------- = - --------- (30) dp[theta] dp[theta] gives the condition of steady precession. 6. _stability of steady motion._ the small oscillations of a conservative system about a configuration of equilibrium, and the criterion of stability, are discussed in mechanics, sec. 23. the question of the stability of given types of motion is more difficult, owing to the want of a sufficiently general, and at the same time precise, definition of what we mean by "stability." a number of definitions which have been propounded by different writers are examined by f. klein and a. sommerfeld in their work _ueber die theorie des kreisels_ (1897-1903). rejecting previous definitions, they base their criterion of stability on the character of the changes produced in the _path_ of the system by small arbitrary disturbing impulses. if the undisturbed path be the _limiting form_ of the disturbed path when the impulses are indefinitely diminished, it is said to be stable, but not otherwise. for instance, the vertical fall of a particle under gravity is reckoned as stable, although for a _given_ impulsive disturbance, however small, the deviation of the particle's position at any time t from the position which it would have occupied in the original motion increases indefinitely with t. even this criterion, as the writers quoted themselves recognize, is not free from ambiguity unless the phrase "limiting form," as applied to a path, be strictly defined. it appears, moreover, that a definition which is analytically precise may not in all cases be easy to reconcile with geometrical prepossessions. thus a particle moving in a circle about a centre of force varying inversely as the cube of the distance will if slightly disturbed either fall into the centre, or recede to infinity, after describing in either case a spiral with an infinite number of convolutions. each of these spirals has, analytically, the circle as its limiting form, although the motion in the circle is most naturally described as unstable. a special form of the problem, of great interest, presents itself in the steady motion of a gyrostatic system, when the non-eliminated co-ordinates q1, q2, ... q_m all vanish (see sec. 5). this has been discussed by routh, lord kelvin and tait, and poincare. these writers treat the question, by an extension of lagrange's method, as a problem of small oscillations. whether we adopt the notion of stability which this implies, or take up the position of klein and sommerfeld, there is no difficulty in showing that stability is ensured if v + [kappa] be a minimum as regards variations of q1, q2, ... q_m. the proof is the same as that of dirichlet for the case of statical stability. we can illustrate this condition from the case of the top, where, in our previous notation, ( mu - [nu]cos [theta]) squared [nu] squared v + [kappa] = mgh cos[theta] + ---------------------- + -----. (1) 2a sin squared [theta] 2c to examine whether the steady motion with the centre of gravity vertically above the pivot is stable, we must put mu = [nu]. we then find without difficulty that v + [kappa] is a minimum provided [nu] squared [>=] 4amgh. the method of small oscillations gave us the condition [nu] squared > 4amgh, and indicated instability in the cases [nu] squared [=<] 4amgh. the present criterion can also be applied to show that the steady precessional motions in which the axis has a constant inclination to the vertical are stable. the question remains, as before, whether it is _essential_ for stability that v + [kappa] should be a minimum. it appears that from the point of view of the theory of small oscillations it is not essential, and that there may even be stability when v + [kappa] is a maximum. the precise conditions, which are of a somewhat elaborate character, have been formulated by routh. an important distinction has, however, been established by thomson and tait, and by poincare, between what we may call _ordinary_ or _temporary_ stability (which is stability in the above sense) and _permanent_ or _secular_ stability, which means stability when regard is had to possible dissipative forces called into play whenever the co-ordinates q1, q2, ... q_m vary. since the total energy of the system at any instant is given (in the notation of sec. 5) by an expression of the form [@] + v + [kappa], where [@] cannot be negative, the argument of thomson and tait, given under mechanics, sec. 23, for the statical question, shows that it is a necessary as well as a sufficient condition for secular stability that v + [kappa] should be a minimum. when a system is "ordinarily" stable, but "secularly" unstable, the operation of the frictional forces is to induce a gradual increase in the amplitude of the free vibrations which are called into play by accidental disturbances. there is a similar theory in relation to the constrained systems considered in sec. 3 above. the equation (21) there given leads to the conclusion that for secular stability of any type of motion in which the velocities q`1, q`2, ... q`_n are zero it is necessary and sufficient that the function v - [tau]0 should be a minimum. the simplest possible example of this is the case of a particle at the lowest point of a smooth spherical bowl which rotates with constant angular velocity ([omega]) about the vertical diameter. this position obviously possesses "ordinary" stability. if a be the radius of the bowl, and [theta] denote angular distance from the lowest point, we have v - [tau]0 = mga(1 - cos [theta]) - 1/2m[omega] squareda squared sin squared [theta]; (2) this is a minimum for [theta] = 0 only so long as [omega] squared < g/a. for greater values of [omega] the only position of "permanent" stability is that in which the particle rotates with the bowl at an angular distance cos^(-1) (g/[omega] squareda) from the lowest point. to examine the motion in the neighbourhood of the lowest point, when frictional forces are taken into account, we may take fixed ones, in a horizontal plane, through the lowest point. assuming that the friction varies as the relative velocity, we have x" = -p squaredx - k(x` + [omega]y), \ (3) y" = -p squaredy - k(y` - [omega]x), / where p squared = g/a. these combine into z" + kz` + (p squared - ik[omega])z = 0, (4) where z = x + iy, i = [root]-1. assuming z = ce^([lambda]t), we find [lambda] = -1/2k(1 [-+] [omega]/p) +- ip, (5) if the square of k be neglected. the complete solution is then x + iy = c1e^(-ss1t)e^(ipt) + c2e^(-ss2t)e^(-ipt), (6) where ss1 = 1/2k(1 - [omega]/p), ss2 = 1/2k(1 + [omega]/p). (7) this represents two superposed circular vibrations, in opposite directions, of period 2[pi]/p. if [omega] < p, the amplitude of each of these diminishes asymptotically to zero, and the position x = 0, y = 0 is permanently stable. but if [omega] > p the amplitude of that circular vibration which agrees in sense with the rotation [omega] will continually increase, and the particle will work its way in an ever-widening spiral path towards the eccentric position of secular stability. if the bowl be not spherical but ellipsoidal, the vertical diameter being a principal axis, it may easily be shown that the lowest position is permanently stable only so long as the period of the rotation is longer than that of the slower of the two normal modes in the absence of rotation (see mechanics, sec. 13). 7. _principle of least action._ stationary action. the preceding theories give us statements applicable to the system at any one instant of its motion. we now come to a series of theorems relating to the whole motion of the system between any two configurations through which it passes, viz. we consider the actual motion and compare it with other imaginable motions, differing infinitely little from it, between the same two configurations. we use the symbol [delta] to denote the transition from the actual to any one of the hypothetical motions. the best-known theorem of this class is that of _least action_, originated by p.l.m. de maupertuis, but first put in a definite form by lagrange. the "action" of a single particle in passing from one position to another is the space-integral of the momentum, or the time-integral of the _vis viva_. the action of a dynamical system is the sum of the actions of its constituent particles, and is accordingly given by the formula _ _ _ / / / a = [sigma] | mvds = [sigma] | mv squareddt = 2 | [tau]dt. (1) _/ _/ _/ the theorem referred to asserts that the free motion of a conservative system between any two given configurations is characterized by the property [delta]a = 0, (2) provided the total energy have the same constant value in the varied motion as in the actual motion. if t, t' be the times of passing through the initial and final configurations respectively, we have _ / t' [delta]a = [delta] | [sigma]m(x` squared + y` squared + z` squared)dt _/t _ / t' = 2 | [delta][tau]dt + 2[tau]'[delta]t' + 2[tau][delta]t, (3) _/t since the upper and lower limits of the integral must both be regarded as variable. this may be written _ _ / t' / t' [delta]a = | [delta][tau]dt + | [sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`)dt + 2[tau]'[delta]t' - 2[tau][delta]t _/t _/t _ _ _ / t' | | t' = | [delta][tau]dt + | [sigma]m (x`[delta]x + y`[delta]y + z`[delta]z | _/t |_ _| t _ / t' - | [sigma]m(x"[delta]x + y"[delta]y + z"[delta]z)dt + 2[tau]'[delta]t' - 2[tau][delta]t. (4) _/t now, by d'alembert's principle, [sigma]m( x"[delta]x + y"[delta]y + z"[delta]z ) = -[delta]v, (5) and by hypothesis we have [delta]([tau] + v) = 0. (6) the formula therefore reduces to _ _ | |t' [delta]a = | [sigma]m (x`[delta]x + y`[delta]y + z`[delta]z) | + 2[tau]'[delta]t' - 2[tau][delta]t. (7) |_ _|t since the terminal configurations are unaltered, we must have at the lower limit [delta]x + x`[delta]t = 0, [delta]y + y`[delta]t = 0, [delta]z + z`[delta]t = 0, (8) with similar relations at the upper limit. these reduce (7) to the form (2). the equation (2), it is to be noticed, merely expresses that the variation of a vanishes _to the first order_; the phrase _stationary action_ has therefore been suggested as indicating more accurately what has been proved. the action in the free path between two given configurations is in fact not invariably a minimum, and even when a minimum it need not be the _least possible_ subject to the given conditions. simple illustrations are furnished by the case of a single particle. a particle moving on a smooth surface, and free from extraneous force, will have its velocity constant; hence the theorem in this case resolves itself into _ / [delta] | ds = 0, (9) _/ i.e. the path must be a geodesic line. now a geodesic is not necessarily the _shortest_ path between two given points on it; for example, on the sphere a great-circle arc ceases to be the shortest path between its extremities when it exceeds 180 deg.. more generally, taking any surface, let a point p, starting from o, move along a geodesic; this geodesic will be a minimum path from o to p until p passes through a point o' (if such exist), which is the intersection with a consecutive geodesic through o. after this point the minimum property ceases. on an anticlastic surface two geodesics cannot intersect more than once, and each geodesic is therefore a minimum path between any two of its points. these illustrations are due to k.g.j. jacobi, who has also formulated the general criterion, applicable to all dynamical systems, as follows:--let o and p denote any two configurations on a natural path of the system. if this be the sole free path from o to p with the prescribed amount of energy, the action from o to p is a minimum. but if there be several distinct paths, let p vary from coincidence with o along the first-named path; the action will then cease to be a minimum when a configuration o' is reached such that two of the possible paths from o to o' coincide. for instance, if o and p be positions on the parabolic path of a projectile under gravity, there will be a second path (with the same energy and therefore the same velocity of projection from o), these two paths coinciding when p is at the other extremity (o', say) of the focal chord through o. the action from o to p will therefore be a minimum for all positions of p short of o'. two configurations such as o and o' in the general statement are called conjugate _kinetic foci_. cf. variations, calculus of. before leaving this topic the connexion of the principle of stationary action with a well-known theorem of optics may be noticed. for the motion of a particle in a conservative field of force the principle takes the form _ / [delta] | vds = 0. (10) _/ on the corpuscular theory of light v is proportional to the refractive index mu of the medium, whence _ / [delta] | muds = 0. (11) _/ hamiltonian principle. in the formula (2) the energy in the hypothetical motion is prescribed, whilst the time of transit from the initial to the final configuration is variable. in another and generally more convenient theorem, due to hamilton, the time of transit is prescribed to be the same as in the actual motion, whilst the energy may be different and need not (indeed) be constant. under these conditions we have _ /t' [delta] | ([tau] - v)dt = 0, (12) _/t where t, t' are the prescribed times of passing through the given initial and final configurations. the proof of (12) is simple; we have _ _ _ /t' /t' /t' [delta] | ([tau] - v)dt = | ([delta][tau] - [delta]v)dt = | {[sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`) - [delta]v}dt _/t _/t _/t _ _ | |t' = | [sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) | |_ _|t _ /t' - | {[sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) + [delta]v}dt (13) _/t the integrated terms vanish at both limits, since by hypothesis the configurations at these instants are fixed; and the terms under the integral sign vanish by d'alembert's principle. the fact that in (12) the variation does not affect the time of transit renders the formula easy of application in any system of co-ordinates. thus, to deduce lagrange's equations, we have _ _ /t' /t'/dp[tau] dp[tau] dpv \ | ([delta][tau]-[delta]v)dt = | ( -------[delta]q`1 + -------[delta]q1 + ... - ----[delta]q1 - ...)dt _/t _/t \ dpq`1 dpq1 dpq1 / _ _ | |t' = | p1[delta]q1 + p2[delta]q2 + ... | |_ _|t _ _ _ /t'| / dp[tau] dpv \ / dp[tau] dpv\ | - | | (p`1 - ------- + ---- )[delta]q1 + (p`2 - ------- + -----)[delta]q2 + ...|dt. (14) _/t |_ \ dpq1 dpq1/ \ dpq2 dpq2/ _| the integrated terms vanish at both limits; and in order that the remainder of the right-hand member may vanish it is necessary that the coefficients of [delta]q1, [delta]q2, ... under the integral sign should vanish for all values of t, since the variations in question are independent, and subject only to the condition of vanishing at the limits of integration. we are thus led to lagrange's equation of motion for a conservative system. it appears that the formula (12) is a convenient as well as a compact embodiment of the whole of ordinary dynamics. extension to cyclic systems. the modification of the hamiltonian principle appropriate to the case of cyclic systems has been given by j. larmor. if we write, as in sec. 1 (25), r = t - [kappa][chi]` - [kappa]'[chi]`' - [kappa]''[chi]`" - ..., (15) we shall have _ /t' [delta] | (r - v)dt = 0, (16) _/t provided that the variation does not affect the cyclic momenta [kappa], [kappa]', [kappa]", ..., and that the configurations at times t and t' are unaltered, so far as they depend on the palpable co-ordinates q1, q2, ... q_m. the initial and final values of the ignored co-ordinates will in general be affected. to prove (16) we have, on the above understandings, _ _ /t' /t' [delta] | (r - v)dt = | ([delta][tau] - [kappa][delta][chi]` - ... -[delta]v)dt _/t _/t _ /t' /dp[tau] dp[tau] \ = | ( -------[delta]q`1 + ... + -------[delta]q1 + ... - [delta]v )dt, (17) _/t \ dpq`1 dpq1 / where terms have been cancelled in virtue of sec. 5 (2). the last member of (17) represents a variation of the integral _ /t' | ([tau] - v)dt _/t on the supposition that [delta]x = 0, [delta]x' = 0, [delta]x" = 0, ... throughout, whilst [delta]q1, [delta]q2, [delta]q_m vanish at times t and t'; i.e. it is a variation in which the initial and final configurations are absolutely unaltered. it therefore vanishes as a consequence of the hamiltonian principle in its original form. larmor has also given the corresponding form of the principle of least action. he shows that if we write _ / a = |(2[tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ...)dt, (18) _/ then [delta]a = 0, (19) provided the varied motion takes place with the same constant value of the energy, and with the same constant cyclic momenta, between the same two configurations, these being regarded as defined by the palpable co-ordinates alone. sec. 8. _hamilton's principal and characteristic functions._ principal function. in the investigations next to be described a more extended meaning is given to the symbol [delta]. we will, in the first instance, denote by it an infinitesimal variation of the most general kind, affecting not merely the values of the co-ordinates at any instant, but also the initial and final configurations and the times of passing through them. if we put _ /t' s = | (t - v)dt, (1) _/t we have, then, _ /t' [delta]s = (t' - v')[delta]t' - (t - v)[delta]t + | ([delta]t - [delta]v)dt _/t _ _ | |t' = (t' - v')[delta]t' - (t - v)[delta]t + |[sigma]m(x`[delta]x + y`[delta]y + z`[delta]z)| (2) |_ _|t let us now denote by x' + [delta]x', y' + [delta]y', z' + [delta]z', the final co-ordinates (i.e. at time t' + [delta]t') of a particle m. in the terms in (2) which relate to the upper limit we must therefore write [delta]x' - x`'[delta]t', [delta]y' - y`'[delta]t', [delta]z' - z`'[delta]t' for [delta]x, [delta]y, [delta]z. with a similar modification at the lower limit, we obtain [delta]s = - h[delta][tau] + [sigma]m(x`'[delta]x' + y`'[delta]y' + z`'[delta]z') - [sigma]m(x`[delta]x + y`[delta]y + z`[delta]z), (3) where h(= t + v) is the constant value of the energy in the free motion of the system, and [tau](= t' - t) is the time of transit. in generalized co-ordinates this takes the form [delta]s = - h[delta][tau] + p'1[delta]q'1 + p'2[delta]q'2 + ... - p1[delta]q1 - p2[delta]q2 - .... (4) now if we select any two arbitrary configurations as initial and final, it is evident that we can in general (by suitable initial velocities or impulses) start the system so that it will of itself pass from the first to the second in any prescribed time [tau]. on this view of the matter, s will be a function of the initial and final co-ordinates (q1, q2, ... and q'1, q'2, ...) and the time [tau], as independent variables. and we obtain at once from (4) dps dps \ p'1 = -----, p'2 = -----, ..., | dpq'1 dpq'2 | > (5) dps dps | p1 = - ----, p2 = - ----, ..., | dpq1 dpq2 / dps and h = - -------. (6) dp[tau] s is called by hamilton the _principal function_; if its general form for any system can be found, the preceding equations suffice to determine the motion resulting from any given conditions. if we substitute the values of p1, p2, ... and h from (5) and (6) in the expression for the kinetic energy in the form [tau]' (see sec. 1), the equation ts + v = h (7) becomes a partial differential equation to be satisfied by s. it has been shown by jacobi that the dynamical problem resolves itself into obtaining a "complete" solution of this equation, involving n + 1 arbitrary constants. this aspect of the subject, as a problem in partial differential equations, has received great attention at the hands of mathematicians, but must be passed over here. characteristic function. there is a similar theory for the function _ / a = 2 | tdt = s + h[tau] (8) _/ it follows from (4) that [delta]a = [tau][delta]h + p'1[delta]q'1 + p'2[delta]q'2 + ... - p1[delta]q1 - p2[delta]q2 - .... (9) this formula (it may be remarked) contains the principle of "least action" as a particular case. selecting, as before, any two arbitrary configurations, it is in general possible to start the system from one of these, with a prescribed value of the total energy h, so that it shall pass through the other. hence, regarding a as a function of the initial and final co-ordinates and the energy, we find dpa dpa \ p'1 = -----, p'2 = -----, ..., | dpq'1 dpq'2 | > (10) dpa dpa | p1 = - ----, p2 = - ----, ..., | dpq1 dpq2 / dpa and [tau] = --- (11) dph a is called by hamilton the _characteristic function_; it represents, of course, the "action" of the system in the free motion (with prescribed energy) between the two configurations. like s, it satisfies a partial differential equation, obtained by substitution from (10) in (7). the preceding theorems are easily adapted to the case of cyclic systems. we have only to write _ _ /t' /t' s = | (r - v)dt= | (t - [kappa][chi]` - [kappa]'[chi]`' - ... - v)dt (12) _/t _/t in place of (1), and _ / a = | (2t - [kappa][chi]` - [kappa]'[chi]`' - ...)dt, (3) _/ in place of (8); cf. sec. 7 ad fin. it is understood, of course, that in (12) s is regarded as a function of the initial and final values of the palpable co-ordinates q1, q2, ... q_m, and of the time of transit [tau], the cyclic momenta being invariable. similarly in (13), a is regarded as a function of the initial and final values of q1, q2, ... q_m, and of the total energy h, with the cyclic momenta invariable. it will be found that the forms of (4) and (9) will be conserved, provided the variations [delta]q1, [delta]q2, ... be understood to refer to the palpable co-ordinates alone. it follows that the equations (5), (6) and (10), (11) will still hold under the new meanings of the symbols. 9. _reciprocal properties of direct and reversed motions._ lagrange's formula. we may employ hamilton's principal function to prove a very remarkable formula connecting any _two_ slightly disturbed natural motions of the system. if we use the symbols [delta] and [delta] to denote the corresponding variations, the theorem is d --[sigma]([delta]p_r.[delta]q_r - [delta]p_r.[delta]q_r) = 0; (1) dt or integrating from t to t', [sigma]([delta]p'_r.[delta]q'_r - [delta]q'_r.[delta]q'_r) = [sigma]([delta]p_r.[delta]q_r - [delta]p_r.[delta]q_r). (2) if for shortness we write dp squareds dp squareds (r,s) = ----------, (r,s') = -----------, (3) dpq_rdpq_s dpq_rdpq'_s we have dpp_r = - [sigma]_s(r,s)[delta]q_s - [sigma]_s(r,s')[delta]q'_s (4) with a similar expression for [delta]p_r. hence the right-hand side of (2) becomes - [sigma]_r{[sigma]_s(r,s)[delta]q_s + [sigma]_s(r,s')[delta]q'_s}[delta]q_r + [sigma]_r{[sigma]_s(r,s)[delta]q_s + [sigma]_s(r,s')[delta]q'_s}[delta]q_r = [sigma]_r[sigma]_s(r,s'){[delta]q_r.[delta]q'_s - [delta]q_r.[delta]q'_s}. (5) the same value is obtained in like manner for the expression on the left hand of (2); hence the theorem, which, in the form (1), is due to lagrange, and was employed by him as the basis of his method of treating the dynamical theory of _variation of arbitrary constants_. helmholtz's reciprocal theorems. the formula (2) leads at once to some remarkable reciprocal relations which were first expressed, in their complete form, by helmholtz. consider any natural motion of a conservative system between two configurations o and o' through which it passes at times t and t' respectively, and let t' - t = [tau]. as the system is passing through o let a small impulse [delta]p_r be given to it, and let the consequent alteration in the co-ordinate q_s after the time [tau] be [delta]q'_s. next consider the _reversed_ motion of the system, in which it would, if undisturbed, pass from o' to o in the same time [tau]. let a small impulse [delta]p'_s be applied as the system is passing through o', and let the consequent change in the co-ordinate q_r after a time [tau] be [delta]q_r. helmholtz's first theorem is to the effect that [delta]q_r : [delta]p'_s = [delta]q'_s : [delta]p_r. (6) to prove this, suppose, in (2), that all the [delta]q vanish, and likewise all the [delta]p with the exception of [delta]p_r. further, suppose all the [delta]q' to vanish, and likewise all the [delta]p' except [delta]p'_s, the formula then gives [delta]p_r.[delta]q_r = - [delta]p'_s.[delta]q'_s, (7) which is equivalent to helmholtz's result, since we may suppose the symbol [delta] to refer to the reversed motion, provided we change the signs of the [delta]p. in the most general motion of a top (mechanics, sec. 22), suppose that a small impulsive couple about the vertical produces after a time [tau] a change [delta][theta] in the inclination of the axis, the theorem asserts that in the reversed motion an equal impulsive couple in the plane of [theta] will produce after a time [tau] a change [delta][psi], in the azimuth of the axis, which is equal to [delta][theta]. it is understood, of course, that the couples have no components (in the generalized sense) except of the types indicated; for instance, they may consist in each case of a force applied to the top at a point of the axis, and of the accompanying reaction at the pivot. again, in the corpuscular theory of light let o, o' be any two points on the axis of a symmetrical optical combination, and let v, v' be the corresponding velocities of light. at o let a small impulse be applied perpendicular to the axis so as to produce an angular deflection [delta][theta], and let ss' be the corresponding lateral deviation at o'. in like manner in the reversed motion, let a small deflection [delta][theta]' at o' produce a lateral deviation ss at o. the theorem (6) asserts that ss ss' ----------------- = ---------------, (8) v'[delta][theta]' v[delta][theta] or, in optical language, the "apparent distance" of o from o' is to that of o' from o in the ratio of the refractive indices at o' and o respectively. helmholtz's second reciprocal theorem. in the second reciprocal theorem of helmholtz the configuration o is slightly varied by a change [delta]q_r in one of the co-ordinates, the momenta being all unaltered, and [delta]q'_s is the consequent variation in one of the momenta after time [tau]. similarly in the reversed motion a change [delta]p'_s produces after time [tau] a change of momentum [delta]p_r. the theorem asserts that [delta]p'_s : [delta]q_r = [delta]p_r : [delta]q'_s (9) this follows at once from (2) if we imagine all the [delta]p to vanish, and likewise all the [delta]q save [delta]q_r, and if (further) we imagine all the [delta]p' to vanish, and all the [delta]q' save [delta]q'_s. reverting to the optical illustration, if f, f', be principal foci, we can infer that the convergence at f' of a parallel beam from f is to the convergence at f of a parallel beam from f' in the inverse ratio of the refractive indices at f' and f. this is equivalent to gauss's relation between the two principal focal lengths of an optical instrument. it may be obtained otherwise as a particular case of (8). we have by no means exhausted the inferences to be drawn from lagrange's formula. it may be noted that (6) includes as particular cases various important reciprocal relations in optics and acoustics formulated by r.j.e. clausius, helmholtz, thomson (lord kelvin) and tait, and lord rayleigh. in applying the theorem care must be taken that in the reversed motion the reversal is complete, and extends to every velocity in the system; in particular, in a cyclic system the cyclic motions must be imagined to be reversed with the rest. conspicuous instances of the failure of the theorem through incomplete reversal are afforded by the propagation of sound in a wind and the propagation of light in a magnetic medium. it may be worth while to point out, however, that there is no such limitation to the use of lagrange's formula (1). in applying it to cyclic systems, it is convenient to introduce conditions already laid down, viz. that the co-ordinates q_r are the palpable co-ordinates and that the cyclic momenta are invariable. special inference can then be drawn as before, but the interpretation cannot be expressed so neatly owing to the non-reversibility of the motion. authorities.--the most important and most accessible early authorities are j.l. lagrange, _mecanique analytique_ (1st ed. paris, 1788, 2nd ed. paris, 1811; reprinted in _oeuvres_, vols. xi., xii., paris, 1888-89); hamilton, "on a general method in dynamics," _phil. trans._ 1834 and 1835; c.g.j. jacobi, _vorlesungen ueber dynamik_ (berlin, 1866, reprinted in _werke_, supp.-bd., berlin, 1884). an account of the extensive literature on the differential equations of dynamics and on the theory of variation of parameters is given by a. cayley, "report on theoretical dynamics," _brit. assn. rep._ (1857), _mathematical papers_, vol. iii. (cambridge, 1890). for the modern developments reference may be made to thomson and tait, _natural philosophy_ (1st ed. oxford, 1867, 2nd ed. cambridge, 1879); lord rayleigh, _theory of sound_, vol. i. (1st ed. london, 1877; 2nd ed. london, 1894); e.j. routh, _stability of motion_ (london, 1877), and _rigid dynamics_ (4th ed. london, 1884); h. helmholtz, "ueber die physikalische bedeutung des prinzips der kleinsten action," _crelle_, vol. c., 1886, reprinted (with other cognate papers) in _wiss. abh._ vol. iii. (leipzig, 1895); j. larmor, "on least action," _proc. lond. math. soc._ vol. xv. (1884); e.t. whittaker, _analytical dynamics_ (cambridge, 1904). as to the question of stability, reference may be made to h. poincare, "sur l'equilibre d'une masse fluide animee d'un mouvement de rotation" _acta math._ vol. vii. (1885); f. klein and a. sommerfeld, _theorie des kreisels_, pts. 1, 2 (leipzig, 1897-1898); a. lioupanoff and j. hadamard, _liouville_, 5me serie, vol. iii. (1897); t.j.i. bromwich, proc. lond. math. soc. vol. xxxiii. (1901). a remarkable interpretation of various dynamical principles is given by h. hertz in his posthumous work _die prinzipien der mechanik_ (leipzig, 1894), of which an english translation appeared in 1900. (h. lb.) dynamite (gr. [greek: dynamis], power), the name given to several explosive preparations containing nitroglycerin (q.v.) which are almost exclusively used for blasting purposes. the first practical application of nitroglycerin in this way was made by a. nobel in 1863. he soaked gunpowder with the liquid and fired the gunpowder by an ordinary fuse. later he found that nitroglycerin could be detonated by the explosion of several materials such as fulminate of mercury, the use of which as a detonator he patented in 1867. in 1866-1867 he experimented with charcoal and other substances, and found the infusorial earth known as kieselguhr, which consists mainly of silica (nearly 95%), eminently adapted to the purpose, as it was inert, non-combustible, and after a little heating and preparation very porous, retaining a large amount of nitroglycerin as water is held in a sponge, without very serious exudation on standing. this kieselguhr dynamite is generally made by incorporating three parts of nitroglycerin with one part of the dry earth, the paste being then formed into cylindrical cartridges. this work is done by hand. generally a small percentage of the kieselguhr is replaced by a mixture containing sodium and ammonium carbonates, talc and ochre. this product is known as dynamite no. 1. disabilities attaching to kieselguhr dynamite are that when placed in water the nitroglycerin is liable to be exuded or displaced, also that, like nitroglycerin itself, it freezes fairly easily and thawing the frozen cartridges is a dangerous operation. other substances, e.g. kaolin, tripoli, magnesia alba (magnesium carbonate), alumina, sugar, charcoal, some powdered salts and mixtures of sawdust and salts, have been shown to be absorbents more or less adapted to the purpose of making a dynamite. charcoal from cork is said to absorb about 90% of its weight of nitroglycerin. with the idea of obtaining greater safety, mixtures have been made of nitroglycerin with wood fibre, charcoal and metallic nitrates. lithofracteur, for instance, consists of 50% nitroglycerin and a mixture of prepared sawdust, kieselguhr and barium nitrate. carbonite contains 25% of nitroglycerin, the remainder being a mixture of wood-meal and alkali nitrates, with about 1% of sulphur. dualin, atlas dynamite and potentite are other modifications. a convenient form in which nitroglycerin can be made up for blasting purposes, especially in wet ground, is the gelatinous material obtained by the action of nitroglycerin, either alone or with the help of solvents, on low-grade or soluble gun-cottons. it is known as blasting gelatin, and was first made by nobel by incorporating 6 or 7% of low nitrated cellulose (collodion cotton or soluble gun-cotton) with slightly warmed nitroglycerin. the result is a transparent plastic material, of specific gravity 1.5 to 1.6, which may be kept under water for a long time without appreciable change. it is less sensitive to detonation than ordinary dynamite, and although its explosion is slightly slower it is more powerful than dynamite and much superior to the liquid nitroglycerin. blasting gelatin also freezes and is sensitive to percussion in this state. camphor and other substances have been added to blasting gelatin to render it more solid and less sensitive. some modifications of blasting gelatin, e.g. gelignite, contain wood-meal and such oxygen-containing salts as potassium nitrate. experience has conclusively shown that dynamites are more satisfactory, quicker, and more intense in action than liquid nitroglycerin. to prevent nitroglycerin and some of the forms of dynamite from freezing it has been proposed to add to them small quantities of either monochlor-dinitroglycerin or of a nitrated poly-glycerin. the former is obtained by first acting upon glycerin with hydrogen chloride to produce _u-_chlorhydrin or chlor-propylene glycol, c3h7o2cl, which is then nitrated as in the case of glycerin. the latter is obtained by heating glycerin for six or seven hours to about 300 deg. c., whereby water is split off in such manner that a diglycerin c6h14o5, for the most part, results. this on nitration in the usual manner gives a product c6h{10}n4o{13}, which burns and explodes in a similar manner to ordinary nitroglycerin, but is less sensitive and does not so easily freeze. the mono- and di-nitrates of glycerin have also been proposed as additions to ordinary nitroglycerin (q.v.) for the same purpose. (w. r. e. h.) dynamo (a shortened form of "dynamo-electric machine," from gr. [greek: dynamis], power), a machine for converting mechanical into electrical energy. the dynamo ranks with the telegraph and telephone as one of the three striking applications of electrical and magnetic science to which the material progress that marked the second half of the 19th century was in no small measure due. since the discovery of the principle of the dynamo by faraday in 1831 the simple model which he first constructed has been gradually developed into the machines of 5000 horse-power or more which are now built to meet the needs of large cities for electric lighting and power, while at the same time the numbers of dynamos in use have increased almost beyond estimate. yet such was the insight of faraday into the fundamental nature of the dynamo that the theory of its action which he laid down has remained essentially unchanged. his experiments on the current which was set up in a coil of wire during its movement across the poles of a magnet led naturally to the explanation of induced electromotive force as caused by the linking or unlinking of magnetic lines of flux with an electric circuit. for the more definite case of the dynamo, however, we may, with faraday, make the transition from line-linkage to the equivalent conception of "line-cutting" as the source of e.m.f.--in other words, to the idea of electric conductors "cutting" or intersecting[1] the lines of flux in virtue of relative motion of the magnetic field and electric circuit. on the 28th of october 1831 faraday mounted a copper disk so that it could be rotated edgewise between the poles of a permanent horse-shoe magnet. when so rotated, it cut the lines of flux which passed transversely through its lower half, and by means of two rubbing contacts, one on its periphery and the other on its spindle, the circuit was closed through a galvanometer, which indicated the passage of a continuous current so long as the disk was rotated (fig. 1). thus by the invention of the first dynamo faraday proved his idea that the e.m.f. induced through the interaction of a magnetic field and an electric circuit was due to the passage of a portion of the electric circuit _across_ the lines of flux, or vice versa, and so could be maintained if the cutting of the lines were made continuous.[2] in comparison with faraday's results, the subsequent advance is to be regarded as a progressive perfecting of the mechanical and electro-magnetic design, partly from the theoretical and partly from the practical side, rather than as modifying or adding to the idea which was originally present in his mind, and of which he already saw the possibilities. [illustration: fig. 1.] a dynamo, then, is a machine in which, by means of continuous relative motion, an electrical conductor or system of conductors forming part of a circuit is caused to cut the lines of a magnetic field or fields; the cutting of the magnetic flux induces an electromotive force in the conductors, and when the circuit is closed a current flows, whereby mechanical energy is converted into electrical energy. little practical use could be made of electrical energy so long as its only known sources were frictional machines and voltaic batteries. the cost of the materials for producing electrical currents on a large scale by chemical action was prohibitive, while the frictional machine only yielded very small currents at extremely high potentials. in the dynamo, on the other hand, electrical energy in a convenient form could be cheaply and easily obtained by mechanical means, and with its invention the application of electricity to a wide range of commercial purposes became economically possible. as a converter of energy from one form to another it is only surpassed in efficiency by another electrical appliance, namely, the transformer (see transformers). in this there is merely conversion of electrical energy at a high potential into electrical energy at a low potential, or vice versa, but in the dynamo the mechanical energy which must be applied to maintain the relative movement of magnetic field and conductor is absorbed, and reappears in an electrical form. a true transformation takes place, and the proportion which the rate of delivery of electrical energy bears to the power absorbed, or in other words the _efficiency_, is the more remarkable. the useful return or "output" at the terminals of a large machine may amount to as much as 95% of the mechanical energy which forms the "input." since it needs some prime mover to drive it, the dynamo has not made any direct addition to our sources of energy, and does not therefore rank with the primary battery or oil-engine, or even the steam-engine, all of which draw their energy more immediately from nature. yet by the aid of the dynamo the power to be derived from waterfalls can be economically and conveniently converted into an electrical form and brought to the neighbouring factory or distant town, to be there reconverted by motors into mechanical power. over any but very short distances energy is most easily transmitted when it is in an electrical form, and turbine-driven dynamos are very largely and successfully employed for such transmission. thus by conducing to the utilization of water-power which may previously have had but little value owing to its disadvantageous situation, the dynamo may almost be said to have added another to our available natural resources. the two essential parts of the dynamo, as required by its definition, may be illustrated by the original disk machine of faraday. they are (1) the _iron magnet_, between the poles of which a magnetic field exists, and (2) the _electrical conductors_, represented by the rotating copper disk. the sector of the disk cutting the lines of the field forms part of a closed electric circuit, and has an e.m.f. induced in it, by reason of which it is no longer simply a conductor, but has become "active." in its more highly developed form the simple copper disk is elaborated into a system of many active wires or bars which form the "winding," and which are so interconnected as to add up their several e.m.f.'s. since these active wires are usually mounted on an iron structure, which may be likened to the keeper or "armature" of a magnet rotating between its poles, the term "armature" has been extended to cover not only the iron core, but also the wires on it, and when there is no iron core it is even applied to the copper conductors themselves. in the dynamo of faraday the "armature" was the rotating portion, and such is the case with modern continuous-current dynamos; in alternators, however, the magnet, or a portion of it, is more commonly rotated while the armature is stationary. it is in fact immaterial to the action whether the one or the other is moved, or both, so long as their relative motion causes the armature conductors to cut the magnetic flux. as to the ultimate reason why an e.m.f. should be thereby induced, physical science cannot as yet yield any surer knowledge than in the days of faraday.[3] for the engineer, it suffices to know that the e.m.f. of the dynamo is due to the cutting of the magnetic flux by the active wires, and, further, is proportional to the rate at which the lines are cut.[4] [illustration: fig. 2.] the equation of the _electromotive force_ which is required in order to render this statement quantitative must contain three factors, namely, the density of the flux in the air-gap through which the armature conductors move, the active length of these wires, and the speed of their movement. for given values of the first and third factors and a single straight wire moved parallel to itself through a uniform field, the maximum rate of cutting is evidently obtained when the three directions of the lines of the conductor's length and of the relative motion are respectively at right angles to each other, as shown by the three co-ordinate axes of fig. 2. the e.m.f. of the single wire is then e = b_glv x 10^(-8) volts (1) where b_g is the density of the flux within the air-gap expressed in c.g.s. lines per square centimetre, l is the active length of the conductor within the field in centimetres, and v is the velocity of movement in centimetres per second. further, the direction in which the e.m.f. has the above maximum value is along the length of the conductor, its "sense" being determined by the direction of the movement[5] in relation to the direction of the field. the second fundamental equation of the dynamo brings to light its mechanical side, and rests on h.c. oersted's discovery of the interaction of a magnetic field and an electric current. if a straight electric conductor through which a current is passing be so placed in a magnetic field that its length is not parallel to the direction of the lines of flux, it is acted on by a force which will move it, if free, in a definite direction relatively to the magnet; or if the conductor is fixed and the magnet is free, the latter will itself move in the opposite direction. now in the dynamo the active wires are placed so that their length is at right angles to the field; hence when they are rotated and an electric current begins to flow under the e.m.f. which they induce, a mutual force at once arises between the copper conductors and the magnet, and the direction of this force must by lenz's law be opposed to the direction of the movement. thus as soon as the disk of fig. 1 is rotated and its circuit is closed, it experiences a mechanical pull or drag which must be overcome by the force applied to turn the disk. while the magnet must be firmly held so as to remain stationary, the armature must be of such mechanical construction that its wires can be forcibly driven through the magnetic field against the mutual pull. this law of electrodynamic action may be quantitatively stated in an _equation of mechanical force_, analogous to the equation (i.) of electromotive force, which states the law of electromagnetic induction. if a conductor of length l cm., carrying a current c amperes, is immersed in a field of uniform density b_g, and the length of the conductor is at right angles to the direction of the lines, it is acted on by a force f = b_glc x 10^(-1) dynes, (2) and the direction of this force is at right angles to the conductor and to the field. the rate at which electrical energy is developed, when this force is overcome by moving the conductor as a dynamo through the field, is ec = b_glvc x 10^(-8) watts, whence the equality of the mechanical power absorbed and the electrical power developed (as required by the law of the conservation of energy) is easily established. the whole of this power is not, however, available at the terminals of the machine; if r_a be the resistance of the armature in ohms, the passage of the current c_a through the armature conductors causes a drop of pressure of c_ar_a volts, and a corresponding loss of energy in the armature at the rate of c_a squaredr_a watts. as the resistance of the external circuit r_e is lowered, the current c = e_a/(r_e + r_a) is increased. the increase of the current is, however, accompanied by a progressive increase in the loss of energy over the armature, and as this is expended in heating the armature conductors, their temperature may rise so much as to destroy the insulating materials with which they are covered. hence the temperature which the machine may be permitted to attain in its working is of great importance in determining its output, the current which forms one factor therein being primarily limited by the heating which it produces in the armature winding. the lower the resistance of the armature, the less the rise of its temperature for a given current flowing through it; and the reason for the almost universal adoption of copper as the material for the armature conductors is now seen to lie in its high conductivity.[6] since the voltage of the dynamo is the second factor to which its output is proportional, the conditions which render the induced e.m.f. a maximum must evidently be reproduced as far as possible in practice, if the best use is to be made of a given mass of iron and copper. the first problem, therefore, in the construction of the dynamo is the disposition of the wires and field in such a manner that the three directions of field, length of active conductors, and movement are at right angles to one another, and so that the relative motion is continuous. reciprocating motion, such as would be obtained by direct attachment of the conductors to the piston of a steam-engine, has been successfully employed only in the special case of an "oscillator,"[7] producing a small current very rapidly changing in direction. rotary motion is therefore universally adopted, and with this two distinct cases arise. either (a) the active length of the wire is parallel to the axis of rotation, or (b) it is at right angles to it. [illustration: fig. 3.] (a) if a conductor is rotated in the gap between the poles of a horse-shoe magnet, and these poles have plane parallel faces opposing one another as in fig. 3, not only is the density of the flux in the interpolar gap small, but the direction of movement is not always at right angles to the direction of the lines, which for the most part pass straight across from one opposing face to the other. when the conductor is midway between the poles (i.e. either at its highest or lowest point), it is at this instant sliding along the lines and does not cut them, so that its e.m.f. is zero. taking this position as the starting-point, as the conductor moves round, its rate of line-cutting increases to a maximum when it has moved through a right angle and is opposite to the centre of a pole-face (as in fig. 3), from which point onward the rate decreases to zero when it has moved through 180 deg.. each time the conductor crosses a line drawn symmetrically through the gap between the poles and at right angles to the axis of rotation, the e.m.f. along its length is reversed in direction, since the motion relatively to the direction of the field is reversed. if the ends of the active conductor are electrically connected to two collecting rings fixed upon, but insulated from, the shaft, two stationary brushes bb can be pressed on the rings so as to make a sliding contact. an external circuit can then be connected to the brushes, which will form the "terminals" of the machine, the periodically reversed or alternating e.m.f. induced in the active conductor will cause an alternating current to flow through conductor and external circuit, and the simplest form of "alternator" is obtained. if the field cut by the straight conductor is of uniform density, and all the lines pass straight across from one pole-face to the other (both of which assumptions are approximately correct), a curve connecting the instantaneous values of the e.m.f. as ordinates with time or degrees of angular movement as abscissae (as shown at the foot of fig. 3), will, if the speed of rotation be uniform, be a sine curve. if, however, the conductor is mounted on an iron cylinder (fig. 4),[8] a sufficient margin being allowed for mechanical clearance between it and the poles, not only will the reluctance of the magnetic circuit be reduced and the total flux and its density in the air-gap b_g be thereby increased, but the path of the lines will become nearly radial, except at the "fringe" near the edges of the pole-tips; hence the relative directions of the movement and of the lines will be continuously at right angles. the shape of the e.m.f. curve will then be as shown in fig. 4--flat-topped, with rounded corners rapidly sloping down to the zero line. [illustration: fig. 4.] but a single wire cannot thus be made to give more than a few volts, and while dynamos for voltages from 5 to 10 are required for certain purposes, the voltages in common use range from 100 to 10,000. it is therefore necessary to connect a number of such wires in series, so as to form an "armature winding." if several similar conductors are arranged along the length of the iron core parallel to the first (fig. 5), the e.m.f.'s generated in the conductors which at any moment are under the same pole are similarly directed, and are opposite to the directions of the e.m.f.'s in the conductors under the other pole (cf fig. 5 where the dotted and crossed ends of the wires indicate e.m.f.'s directed respectively towards and away from the observer). two distinct methods of winding thence arise, the similarity of the e.m.f.'s under the same pole being taken advantage of in the first, and the opposite e.m.f.'s under n and s poles in the second. [illustration: fig. 5.] [illustration: fig. 6.] 1. the first, or _ring_-winding, was invented by dr antonio pacinotti of florence[9] in 1860, and was subsequently and independently reintroduced in 1870[10] by the belgian electrician, zenobe theophile gramme, whence it is also frequently called the "gramme" winding. by this method the farther end of conductor 1 (fig. 5) is joined in series to the near end of conductor 2; this latter lies next to it on the surface of the core or immediately above it, so that both are simultaneously under the same pole-piece. for this series connexion to be possible, the armature core must be a hollow cylinder, supported from the shaft on an open non-magnetic spider or hub, between the arms of which there is room for the internal wire completing the loop (fig. 6). the end of one complete loop or turn embracing one side of the armature core thus forms the starting-point for another loop, and the process can be continued if required to form a coil of two or more turns. in the ring armature the iron core serves the double purpose of conducting the lines across from one pole to the other, and also of shielding from the magnetic flux the hollow interior through which the connecting wires pass. any lines which leak across the central space are cut by the internal wires, and the direction of cutting is such that the e.m.f. caused thereby opposes the e.m.f. due to the active conductors proper on the external surface. if, however, the section of iron in the core be correctly proportioned, the number of lines which cross the interior will bear but a small ratio to those which pass entirely through the iron, and the counter e.m.f. of the internal wires will become very small; they may then be regarded simply as connectors for joining the external active wires in series. [illustration: fig. 7.] 2. the second or _drum_ method was used in the original "shuttle-wound" armatures invented by dr werner von siemens in 1856, and is sometimes called the "siemens" winding. the farther end of conductor 1 (fig. 5) is joined by a connecting wire to the farther end of another conductor 2' situated nearly diametrically opposite on the other side of the core and under the opposite pole-piece. the near end of the complete loop or turn is then brought across the end of the core, and can be used as the starting-point for another loop beginning with conductor 2, which is situated by the side of the first conductor. the iron core may now be solid from the surface to the shaft, since no connecting wires are brought through the centre, and each loop embraces the entire armature core (fig. 7). by the formation of two loops in the ring armature and of the single loop in the drum armature, two active wires are placed in series; the curves of instantaneous e.m.f. are therefore similar in shape to that of the single wire (fig. 4), but with their ordinates raised throughout to double their former height, as shown at the foot of fig. 6. next, if the free ends of either the ring or drum loops, instead of being connected to two collecting rings, are attached to the two halves of a split-ring insulated from the shaft (as shown in fig. 7 in connexion with a drum armature), and the stationary brushes are so set relatively to the loops that they pass over from the one half of the split-ring to the other half at the moment when the loops are passing the centre of the interpolar gap, and so are giving little or no e.m.f., each brush will always remain either positive or negative. the current in the external circuit attached to the brushes will then have a constant direction, although the e.m.f. in the active wires still remains alternating; the curve of e.m.f. obtained at the brushes is thus (as in fig. 7) entirely above the zero line. the first dynamo of h. pixii,[11] which immediately followed faraday's discovery, gave an alternating current, but in 1832[12] the alternator was converted into a machine giving a _unidirected current_ by the substitution of a rudimentary "commutator" in place of mercury collecting cups. (b) so far the length of the active wires has been parallel to the axis of rotation, but they may equally well be arranged perpendicularly thereto. the poles will then have plane faces and the active wires will be disposed with their length approximately radial to the axis of the shaft. in order to add their e.m.f.'s in series, two types of winding may be employed, which are precisely analogous in principle to the ring and drum windings under arrangement (a). 3. the _discoidal_ or flat-ring armature is equivalent to a ring of which the radial depth greatly exceeds the length, with the poles presented to one side of the ring instead of embracing its cylindrical surface. a similar set of poles is also presented to the opposite side of the ring, like poles being opposite to one another, so that in effect each polar surface is divided into two halves, and the groups of lines from each side bifurcate and pass circumferentially through the armature core to issue into the adjacent poles of opposite sign. 4. in the _disk_ machine, no iron core is necessary for the armature, the two opposite poles of unlike sign being brought close together, leaving but a short path for the lines in the air-gap through which the active wires are rotated. if the above elementary dynamos are compared with fig. 1, it will be found that they all possess a distinctive feature which is not present in the original disk machine of faraday. in the four types of machine above described each active wire in each revolution first cuts the group of lines forming a field in one direction, and then cuts the same lines again in the opposite direction relatively to the sense of the lines, so that along the length of the wire the e.m.f. alternates in direction. but in the dynamo of fig. 1 the sector of the copper disk which is at any moment moving through the magnetic field and which forms the single active element is always cutting the lines in the same manner, so that the e.m.f. generated along its radial length is continuous and unchanged in direction. this radical distinction differentiates the two classes of _heteropolar_ and _homopolar_ dynamos, faraday's disk machine of fig. 1 being the type of the latter class. in it the active element may be arranged either parallel or at right angles to the axis of rotation; but in both cases, in order to increase the e.m.f. by placing two or more elements in series, it becomes necessary either (1) to employ some form of sliding contact by which the current may be collected from the end of one active element and passed round a connecting wire into the next element without again cutting the field in the reverse direction, or (2) to form on the armature a loop of which each side is alternately active and inactive. the first method limits the possibilities of the homopolar machine so greatly when large currents and high voltages are required that it is now only used in rare instances, as e.g. occasionally in dynamos driven by steam-turbines which have a very high speed of rotation. the second alternative may be carried into effect with any of the four methods of armature winding, but is practically confined to the drum and disk types. in its drum form the field is divided into two or more projecting poles, all of the same sign, with intervening neutral spaces of equal width, and the span of the loop in the direction of rotation is at least equal to the width of a polar projection, as in fig. 8, where two polar projections are shown. each side of the loop then plays a dual part; it first cuts the lines of one polar projection and generates an e.m.f., and next becomes an inactive connecting wire, while the action is taken up by the opposite side of the loop which has previously served as a connector but now cuts the lines of the next polar projection. the e.m.f. is thus always in the same direction along the side which is at any moment active, but alternates round the loop as a whole, and the distinctive peculiarity of the homopolar machine, so soon as any form of "winding" is introduced into its armature, is lost. it results that the homopolar principle, which would prima facie appear specially suitable for the generation of a unidirectional e.m.f. and continuous current, can seldom be used for this purpose and is practically confined to alternators. it may therefore be said that in almost all dynamos, whether they supply an alternating or a continuous current in the external circuit, the e.m.f. and current in the armature are alternating. [illustration: fig. 8.] ring winding was largely employed in early continuous-current dynamos and also in the alternators of gramme and h. wilde, and later of auguste de meritens. disk winding was also successfully introduced for alternators, as in the magneto-machines of nollet (1849) and the alternators of wilde (1866) and siemens (1878), and its use was continued in the machines of w.m. mordey and s.z. ferranti. but although the ring, discoidal-ring and disk methods of winding deserve mention from their historical importance, experience has shown that drum winding possesses a marked superiority for both electrical and manufacturing reasons; the three former methods have in fact been practically discarded in its favour, so that the drum method will hereafter alone be considered. the drum coil, composed of several loops wound side by side, may therefore be regarded as the constituent active element out of which the armature winding of the modern dynamo is developed. its application to the multipolar machine is easily followed from fig. 9, which illustrates the heteropolar type of dynamo. the span of the loops, which is nearly 180 deg. or across the diameter of the two-pole machine, is reduced approximately to 90 deg. in the four-pole or to 60 deg. in the six-pole machine and so on, the curvature of the coil becoming gradually less as the number of poles is increased. the passage of a coil through two magnetic fields of opposite direction yields a complete wave of e.m.f., such as is shown in fig. 6, and the time in seconds taken to pass through such a complete cycle is the "period" of the alternating e.m.f. the number of complete periods through which the e.m.f. of the coil passes per second is called the "periodicity" or "frequency" of the machine. in the bipolar machine this is equal to the number of revolutions per second, and in the multipolar machine it is equal to the number of pairs of fields through which the coil passes in one second; hence in general the periodicity is pn/60, where n = the number of revolutions per minute and p = the number of pairs of poles, and this holds true of the e.m.f. and current round the coil, even though the e.m.f. and current furnished to the external circuit may be rendered unidirectional or continuous. the only difference on this point is that in the continuous-current machine the poles are usually fewer than in the alternator, and the periodicity is correspondingly lower. thus in the former case the number of poles ranges from 2 to 12 and the usual frequencies from 5 to 20; but with alternators the frequencies in commercial use range from 25 to 120, and in large machines driven by slow-speed engines the number of poles may even be as high as 96. [illustration: fig. 9. i. smooth. ii. toothed.] [illustration: fig. 10.] the drum coil may be applied either to the external surface of a rotating armature, the field-magnet being external and stationary (fig. 9), or to the internal surface of a stationary armature (fig. 10), the field-magnet being internal and rotating. while the former combination is universally adopted in the continuous-current dynamo, the latter is more usual in the modern alternator. in either case the iron armature core must be "laminated"; the passage of the lines of the field across its surface sets up e.m.f.'s which are in opposite directions under poles of opposite sign, so that if the core were a solid mass a current-sheet would flow along its surface opposite to a pole, and complete its circuit by passing through the deeper layers of metal or by returning in a sheet under a pole of opposite sign. such "eddy-currents" can be practically avoided by dividing the metal core into laminations at right angles to the length of the active wires which are themselves arranged to secure the greatest rate of line-cutting and maximum e.m.f. the production of the eddy-current e.m.f. is not thereby prevented, but the paths of the eddy-currents are so broken up that the comparatively high resistance with which they meet reduces their amount very greatly. the laminae must be lightly insulated from one another, right up to their edges, so that the e.m.f.'s which still act across their thickness will not be added up along the length of the core, but will only produce extremely small currents circulating through the interior of the separate laminations. each thin iron plate is either coated with an insulating varnish or has one of its sides covered with a sheet of very thin paper; the thickness of the laminae is usually about one-fortieth of an inch, and if this is not exceeded the rate at which energy is dissipated by eddy-currents in the core is so far reduced that it does not seriously impair the efficiency of the machine. lastly, the drum coils may be either attached to the surface of a smooth armature core (fig. 9, i.), or may be wound through holes formed close to the periphery of the core, or may be embedded in the slots between projecting iron teeth (figs. 9 [ii.] and 10). originally employed by antonio pacinotti in connexion with ring winding, the toothed armature was after some considerable use largely discarded in favour of the smooth core; it has, however, been reintroduced with a fuller understanding of the special precautions necessitated in its design, and it is now so commonly used that it may be said to have superseded the smooth-surface armature. not only does the toothed armature reduce the length of the air-gap to the minimum permitted by mechanical and magnetic considerations, and furnish better mechanical protection to the armature coils, but it also ensures the positive holding of the active wires against the mechanical drag which they experience as they pass through the magnetic field. further, the active wires in the toothed armature are relieved of a large proportion of this mechanical drag, which is transferred to the iron teeth. the lines of the field, after passing through the air-gap proper, divide between the teeth and the slots in proportion to their relative permeances. hence at any moment the active wires are situated in a weak field, and for a given armature current the force on them is only proportional to this weak field. this important result is connected with the fact that when the armature is giving current the distribution of the lines over the face of each tooth is distorted, so that they become denser on the "trailing" side than on the "leading" side;[13] the effect of the non-uniform distribution acting on all the teeth is to produce a magnetic drag on the armature core proportional to the current passing through the wires, so that the total resisting force remains the same as if the armature had a smooth core. the amount by which the stress on the active wires is reduced entirely depends upon the degree to which the teeth are saturated, but, since the relative permeability of iron even at a flux density of 20,000 lines per sq. cm. is to that of air approximately as 33:1, the embedded wires are very largely relieved of the driving stress. an additional gain is that solid bars of much greater width can be used in the toothed armature than on a smooth core without appreciable loss from eddy-currents within their mass. a disadvantage of the slotted core is, however, that it usually necessitates the lamination of the pole-pieces. if the top of the slot is open, and its width of opening is considerably greater than the length of the air-gap from the iron of the pole-face to the surface of the teeth, the lines become unequally distributed not only at the surface of the teeth, but also at the face of the pole-pieces; and this massing of the lines into bands causes the density at the pole-face to be rhythmically varied as the teeth pass under it. no such variation can take place in a solid mass of metal without the production of eddy-currents within it; hence if the width of the slot-opening is equal to or exceeds twice the length of the single air-gap, lamination of the pole-pieces in the same plane as that of the armature core becomes advisable. if the wires are threaded through holes or tunnels pierced close to the periphery of the core, the same advantages are gained as with open slots, and lamination of the pole-pieces is rendered unnecessary. but on the other hand, the process of winding becomes laborious and expensive, while the increase in the inductance of the coils owing to their being surrounded by a closed iron circuit is prejudicial to sparkless commutation in the continuous-current dynamo and to the regulation of the voltage of the alternator. a compromise is found in the half-closed slot, which is not uncommon in alternators, although the open slot is more usual in continuous-current dynamos. with the addition of more turns to the elementary drum loop or of several complete coils, new questions arise, and in connexion therewith the two great classes of machines, viz. alternators and continuous-current dynamos, which have above been treated side by side, diverge considerably, so that they are best considered separately. the electromotive-force equation of the alternator will be first deduced, and subsequently that of the continuous-current machine. [illustration: fig. 11.] corresponding to the number of pairs of poles in the multipolar alternator, it is evident that there may also be an equal number of coils as shown diagrammatically in fig. 11. the additional coils, being similarly situated in respect to other pairs of poles, will exactly reproduce the e.m.f. of the original coil in phase and magnitude, so that when they are connected in series the total e.m.f. will be proportional to the number of coils in series; or if they are connected in parallel, while not adding to the e.m.f., they will proportionately increase the current-carrying capacity of the combination. but within each coil the addition of more loops will not cause an equal increase in the total e.m.f., unless the phases of the component e.m.f.'s due to the several turns are identical, and on this account it becomes necessary to consider the effect of the width of the coil-side. [illustration: fig. 12.] if the additional loops are wound within the same slots as the original loop, the winding is "concentrated," and each turn will then add the same e.m.f. but if the coil-side is divided between two or more slots, the phase of the e.m.f. yielded by the wires in one slot being different from that of the wires in another neighbouring slot, the sum of all the e.m.f.'s will be less than the e.m.f. of one component loop multiplied by the number of loops or turns in the coil. the percentage reduction in the e.m.f. will depend upon the number of the slots in a coil-side and their distance apart, i.e. on the virtual width of the coil-side expressed as a fraction of the "pole-pitch" or the distance measured along the pitch-line from the centre of one pole to the centre of a neighbouring pole of opposite sign (fig. 12). the winding is now to be regarded as "grouped," since a small number of distinct phases corresponding to the groups within the two, three or four slots have to be compounded together. as the number of slots per coil-side is increased, an approach is gradually made to the case of "uniform distribution," such as would obtain in a smooth-core armature in which the turns of the coil are wound closely side by side. thus in the six-turn coil of fig. 12 a, which represents the development of a two-pole armature when the core is cut down to the shaft and opened out flat, there are in effect six phases compounded together, each of which differs but little from that of its next neighbour. with numerous wires lying still closer together a large number of phases are compounded until the distribution becomes practically uniform; the decrease in the