CONTINENTAL SHELF

continental shelf, the term in physical geography for the submerged platform upon which a continent or island stands in relief. if a coin or medal be partly sunk under water the image and superscription will stand above water and represent a continent with adjacent islands; the sunken part just submerged will represent the continental shelf and the edge of the coin the boundary between it and the surrounding deep, called by professor h. k. h. wagner the continental slope. if the lithosphere surface be divided into three parts, namely, the continent heights, the ocean depths, and the transitional area separating them, it will be found that this transitional area is almost bisected by the coast-line, that nearly one-half of it (10,000,000 sq. m.) lies under water less than 100 fathoms deep, and the remainder 12,000,000 sq. m. is under 600 ft. in elevation. there are thus two continuous plain systems, one above water and one under water, and the second of these is called the continental shelf. it represents the area which would be added to the land surface if the sea fell 600 ft. this shelf varies in width. round africa--except to the south--and off the western coasts of america it scarcely exists. it is wide under the british islands and extends as a continuous platform under the north sea, down the english channel to the south of france; it unites australia to new guinea on the north and to tasmania on the south, connects the malay archipelago along the broad shelf east of china with japan, unites north-western america with asia, sweeps in a symmetrical curve outwards from north-eastern america towards greenland, curving downwards outside newfoundland and holding hudson bay in the centre of a shallow dish. in many places it represents the land planed down by wave action to a plain of marine denudation, where the waves have battered down the cliffs and dragged the material under water. if there were no compensating action in the differential movement of land and sea in the transitional area, the whole of the land would be gradually planed down to a submarine platform, and all the globe would be covered with water. there are, however, periodical warpings of this transitional area by which fresh areas of land are raised above sea-level, and fresh continental coast-lines produced, while the sea tends to sink more deeply into the great ocean basins, so that the continents slowly increase in size. "in many cases it is possible that the continental shelf is the end of a low plain submerged by subsidence; in others a low plain may be an upheaved continental shelf, and probably wave action is only one of the factors at work" (h. r. mill, _realm of nature_, 1897). continued fractions. in mathematics, an expression of the form b2 a1 ± ----------- b3 a2 ± ----------- b4 a3 ± ---------- b5 a4 ± ---- a5 ± ..., where a1, a2, a3, ... and b2, b3, b4, ... are any quantities whatever, positive or negative, is called a "continued fraction." the quantities a1 ..., b2 ... may follow any law whatsoever. if the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a1 ..., b2 ... is infinite it is said to be a _non-terminating_ or _infinite_ continued fraction. if b2/a2, b3/a3 ..., the _component fractions_, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be _recurring_ or _periodic_. it is obvious that every terminating continued fraction reduces to a commensurable number. the notation employed by english writers for the general continued fraction is b2 b3 b4 a1 ± -- -- -- ... a2 ± a3 ± a4 ± continental writers frequently use the notation b2 b3 b4 b2 | b3 | b4 | a1 ± -- ± -- ± -- ± ..., or a1 ± |----| ± |----| ± |----| ± ... a2 a3 a4 | a2 | a3 | a4 the terminating continued fractions b2 b2 b3 b2 b3 b4 a1, a1 + --, a1 + -- --, a1 + -- -- --, ... a2 a2 + a3 a2 + a3 + a4 reduced to the forms a1 a1a2 + b2 a1a2a3 + b2a3 + b2a1 --, ---------, --------------------, 1 a2 a2a3 + b3 a1a2a3a4 + b2a3a4 + b3a1a4 + b4a1a2 + b2b4 ------------------------------------------, ... a2a3a4 + a4b3 + a2b4 are called the successive convergents to the general continued fraction. their numerators are denoted by p1, p2, p3, p4...; their denominators by q1, q2, q3, q4.... we have the relations p_n = a_{n}p_{n-1} + b_{n}p_{n-2}, q_n = a_{n}q_{n-1} + b_{n}q_{n-2}. b2 b3 b4 in the case of the fraction a1 - -- -- -- ..., we have the a2 - a3 - a4 - relations p_n = a_{n}p_{n-1} - b_{n}p_{n-2}, q_n= a_{n}q_{n-1} - b_{n}q_{n-2}. taking the quantities a1 ..., b2 ... to be all positive, a continued b2 b3 fraction of the form a1 + -- -- ... is called a _continued fraction a2 + a3 + b2 b3 b4 of the first class_; a continued fraction of the form -- -- -- ... a2 - a3 - a4 - called a _continued fraction of the second class_. 1 1 1 a continued fraction of the form a1 + -- -- -- ..., where a2 + a3 + a4 + a1, a2, a3, a4 ... are all _positive integers_, is called a _simple continued fraction_. in the case of this fraction a1, a2, a3, a4 ... are called the successive _partial quotients_. it is evident that, in this case, p1, p2, p3 ..., q1, q2, q3 ..., are two series of positive integers increasing without limit if the fraction does not terminate. b2 b3 b4 the general continued fraction a1 + -- -- -- ... is evidently a2 + a3 + a4 + equal, convergent by convergent, to the continued fraction [lambda]2b2 [lambda]2[lambda]3b3 [lambda]3[lambda]4b4 a1 + ----------- -------------------- -------------------- ..., [lambda]2a2 + [lambda]3a3 + [lambda]4a4 + where [lambda]2, [lambda]3, [lambda]4, ... are any quantities whatever, so that by choosing [lambda]2b2 = 1, [lambda]2[lambda]3b3 = 1, &c., it can be reduced to any equivalent continued fraction of the form 1 1 1 a1 + -- -- -- ... d2 + d3 + d4 + _simple continued fractions._ 1. the simple continued fraction is both the most interesting and important kind of continued fraction. any quantity, commensurable or incommensurable, can be expressed uniquely as a simple continued fraction, terminating in the case of a commensurable quantity, non-terminating in the case of an incommensurable quantity. a non-terminating simple continued fraction must be incommensurable. in the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the last 1 partial quotient, a_n as a_n - 1 + --. 1 the numerators and denominators of the successive convergents obey the law p_{n}q_{n-1} - p_{n-1}q_n = (-1)^n, from which it follows at once that every convergent is in its lowest terms. the other principal properties of the convergents are:-- the odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property. every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent. every convergent is nearer to the value of the whole fraction than any preceding convergent. every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent. the difference between the continued fraction and the n^{th} convergent 1 a_{n+2} is less than ------------, and greater than ------------. these limits q_{n}q_{n+1} q_{n}q_{n+2} may be replaced by the following, which, though not so close, are 1 1 simpler, viz. ------- and ------------------ . q^{2}_n q_n(q_n + q_{n+1}) every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since p_n p_{n-1} 1 p_n p_{n-1} --- ~ ------- = ------------, lt. ----- = lt. -------, q_n q_{n-1} q_{n}q_{n-1} q_n q_{n-1} so that its value cannot oscillate. the chief practical use of the simple continued fraction is that by means of it we can obtain rational fractions which approximate to any quantity, and we can also estimate the error of our approximation. thus a continued fraction equivalent to [pi] (the ratio of the circumference to the diameter of a circle) is 1 1 1 1 1 1 3 + - -- -- --- -- -- 7 + 15 + 1 + 292 + 1 + 1 + ... of which the successive convergents are 3 22 333 355 103993 --, --, ---, ---, ------, &c., 1 7 106 113 33102 the fourth of which is accurate to the sixth decimal place, since the error lies between 1/q4q5 or .0000002673 and a6/q4q6 or .0000002665. similarly the continued fraction given by euler as equivalent to 1⁄2(e -1) (e being the base of napierian logarithms), viz. 1 1 1 1 1 -- -- -- -- -- 1 + 6 + 10 + 14 + 18 + ..., may be used to approximate very rapidly to the value of e. for the application of continued fractions to the problem "to find the fraction, whose denominator does not exceed a given integer d, which shall most closely approximate (by excess or defect, as may be assigned) to a given number commensurable or incommensurable," the reader is referred to g. chrystal's _algebra_, where also may be found details of the application of continued fractions to such interesting and important problems as the recurrence of eclipses and the rectification of the calendar (q.v.). lagrange used simple continued fractions to approximate to the solutions of numerical equations; thus, if an equation has a root between two integers a and a + 1, put x = a + 1/y and form the equation in y; if the equation in y has a root between b and b + 1, put y = b + 1/z, and so on. such a method is, however, too tedious, compared with such a method as homer's, to be of any practical value. the solution in integers of the indeterminate equation ax + by = c may be effected by means of continued fractions. if we suppose a/b to be converted into a continued fraction and p/q to be the penultimate convergent, we have aq - bp = +1 or -1, according as the number of convergents is even or odd, which we can take them to be as we please. if we take aq-bp = +1 we have a general solution in integers of ax + by = c, viz. x = cq - bt, y = at - cp; if we take aq - bp = -1, we have x = bt - cq, y = cp - at. an interesting application of continued fractions to establish a unique correspondence between the elements of an aggregate of m dimensions and an aggregate of n dimensions is given by g. cantor in vol. 2 of the _acta mathematica_. applications of simple continued fractions to the theory of numbers, as, for example, to prove the theorem that a divisor of the sum of two squares is itself the sum of two squares, may be found in j. a. serret's _cours d'algebre superieure_. 2. _recurring simple continued fractions._--the infinite continued fraction 1 1 1 1 1 1 1 1 1 1 a1 + -- -- --- -- -- --- -- -- --- -- a2 + a3 ... + a_n + b1 + b2 ... + b_n + b1 + b2 ... + b_n + b1 + ..., where, after the n^{th} partial quotient, the cycle of partial quotients b1, b2, ..., b_n recur in the same order, is the type of a recurring simple continued fraction. the value of such a fraction is the positive root of a quadratic equation whose coefficients are real and of which one root is negative. since the fraction is infinite it cannot be commensurable and therefore its value is a quadratic surd number. conversely every positive quadratic surd number, when expressed as a simple continued fraction, will give rise to a recurring fraction. thus __ 1 1 1 1 1 2 - \/ 3 = -- -- -- -- -- 3 + 1 + 2 + 1 + 2 + ..., ___ 1 1 1 1 1 1 1 1 \/ 28 = 5 + -- -- -- -- -- -- -- -- 3 + 2 + 3 + 10 + 3 + 2 + 3 + 10 + ... the second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd. there is only one non-recurring partial quotient a1. if b1, b2, ..., b_n is the cycle of recurring quotients, then b_n = 2a1, b1 = b_{n-1}, b2 = b_{n-2}, b3 = b_{n-3}, &c. in the case of a recurring continued fraction which represents [sqr]n, where n is an integer, if n is the number of partial quotients in the recurring cycle, and p_{nr}/q_{nr} the nr^{th} convergent, then p^2_{nr} -nq^2_{nr} = (-1)^{nr}, whence, if n is odd, integral solutions of the indeterminate equation x2 - ny2 = ±1 (the so-called pellian equation) can be found. if n is even, solutions of the equation x2 -ny2 = +1 can be found. the theory and development of the simple recurring continued fraction is due to lagrange. for proofs of the theorems here stated and for applications to the more general indeterminate equation x2 -ny2 = h the reader may consult chrystal's _algebra_ or serret's _cours d'algebre superieure_; he may also profitably consult a tract by t. muir, _the expression of a quadratic surd as a continued fraction_ (glasgow, 1874). _the general continued fraction._ 1. _the evaluation of continued fractions._--the numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation u_n = a_{n}u_{n-1} + b_{n}u_{n-2}. when we can solve this equation we have an expression for the n^{th} convergent to the fraction, generally in the form of the quotient of two series, each of n terms. as an example, take the fraction (known as brouncker's fraction, after lord brouncker) 1 12 32 52 72 -- -- -- -- -- 1 + 2 + 2 + 2 + 2 + ... here we have u_{n+1} = 2u_n + (2n-1)2u_{n-1}, whence u_{n+1} - (2n + 1)u_n = -(2n - 1){u_n - (2n - 1)u_{n-1}}, and we readily find that p_n 1 1 1 1 ----- = 1 - -- + -- - -- + ... ± ------, q_n 3 5 7 2n + 1 whence the value of the fraction taken to infinity is 1⁄4[pi]. it is always possible to find the value of the n^{th} convergent to a recurring continued fraction. if r be the number of quotients in the recurring cycle, we can by writing down the relations connecting the successive p's and q's obtain a linear relation connecting p_{nr+m}, p_{(n-1)r+m}, p_{(n-2)r+m}, in which the coefficients are all constants. or we may proceed as follows. (we need not consider a fraction with a non-recurring part). let the fraction be a1 a2 a_r a1 -- -- --- -- b1 + b2 + ... + b_r + b1 + ... p_{nr+m} a1 a2 a_r let u_n = --------; then u_n = -- -- ------------, leading q_{nr+m} b1 + b2 + ... + b_r + u_{n1} to an equation of the form au_{n}u_{n-1} + bu_n + cu_{n-1} + d = 0, where a, b, c, d are independent of n, which is readily solved. 2. _the convergence of infinite continued fractions._--we have seen that the simple infinite continued fraction converges. the infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents. it may, however, oscillate. we have the relation p_{n}q_{n-1} - p_{n-1}q_n = (-1)^{n}b2b3...b_n, p_n p_{n-1} b2b3 ... b_n from which --- - ------- = (-1)^n ------------, and the limit of the q_n q_{n-1} q_{n}q_{n-1} right-hand side is not necessarily zero. the tests for convergency are as follows: let the continued fraction of the first class be reduced to the form 1 1 1 d1 + -- -- -- , then it is convergent if at least one of the d2 + d3 + d4 + ... series d3 + d5 + d7 + ..., d2 + d4 + d6 + ... diverges, and oscillates if both these series converge. for the convergence of the continued fraction of the second class there is no complete criterion. the following theorem covers a large number of important cases. "if in the infinite continued fraction of the second class a_n [>=] b_n + 1 for all values of n, it converges to a finite limit not greater than unity." 3. _the incommensurability of infinite continued fractions._--there is no general test for the incommensurability of the general infinite continued fraction. two cases have been given by legendre as follows:-- if a2, a3, ..., a_n, b2, b3, ...,b_n are all positive integers, then b2 b3 b_{n} i. the infinite continued fraction -- -- ----- converges a2 + a3 + ... + a_{n} + ... to an incommensurable limit if after some finite value of n the condition a_{n} [not <] b_{n} is always satisfied. b2 b3 b_{n} ii. the infinite continued fraction -- -- ----- a2 - a3 - ... - a_{n} - ... converges to an incommensurable limit if after some finite value of n the condition a_{n} [>=] b_{n} + 1 is always satisfied, where the sign > need not always occur but must occur _infinitely often_. _continuants._ the functions p_{n} and q_{n}, regarded as functions of a1, ..., a_{n}, b2, ..., b_{n} determined by the relations p_{n} = a_{n}p_{n-1} + b_{n}p_{n-2}, q_{n} = a_{n}q_{n-1} + b_{n}q_{n-2}, with the conditions p1 = a1, p0 = 1; q2 = a2, q1 = 1, q0 = 0, have been studied under the name of _continuants_. the notation adopted is / b2,...,b_{n}\ p_{n} = k ( ), \a1, a2,...,a_{n}/ and it is evident that we have / b3,...,b_{n}\ q_{n} = k ( ). \a2, a3,...,a_{n}/ the theory of continuants is due in the first place to euler. the reader will find the theory completely treated in chrystal's _algebra_, where will be found the exhibition of a prime number of the form 4p + 1 as the actual sum of two squares by means of continuants, a result given by h. j. s. smith. the continuant / b2, b3, ..., b_{n}\ k ( ) is also equal to the determinant \a1, a2, a3, ..., a_{n}/ is also equal to the determinant | a1 b2 0 0 . . . 0 | | -1 a2 b3 0 . . . 0 | | 0 -1 a3 b4 . . . 0 | | 0 0 -1 a4 b5 . . -- | | | | u -1 a_{n-1} b_{n} | | 0 0 -- -- 0 0 -1 a_{n} |, from which point of view continuants have been treated by w. spottiswoode, j. j. sylvester and t. muir. most of the theorems concerning continued fractions can be thus proved simply from the properties of determinants (see t. muir's _theory of determinants_, chap. iii.). perhaps the earliest appearance in analysis of a continuant in its determinant form occurs in lagrange's investigation of the vibrations of a stretched string (see lord rayleigh, _theory of sound_, vol. i. chap. iv.). _the conversion of series and products into continued fractions._ 1. a continued fraction may always be found whose n^{th} convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product. in fact, a continued fraction b1 b2 b_{n} -- -- ----- can be constructed having for the a1 + a2 + ... + a_{n} + ... numerators of its successive convergents any assigned quantities p1, p2, p3, ..., p_{n}, and for their denominators any assigned quantities q1, q2, q3, ..., q_{n} ... the partial fraction b_{n}/a_{n} corresponding to the n^{th} convergent can be found from the relations p_n = a_{n}p_{n-1} + b_{n}p_{n-2}, q_n = a_{n}q_{n-1} + b_{n}q_{n-2}; and the first two partial quotients are given by b1 = p1, a1 = q1, b1a2 = p2, a1a2 + b2 = q2. if we form then the continued fraction in which p1, p2, p3, ..., p_{n} are u1, u1 + u2, u1 + u2 + u3, ..., u1 + u2 + ..., u_{n}, and q1, q2, q3, ..., q_{n} are all unity, we find the series u1 + u2 + ..., u_{n} equivalent to the continued fraction u1 u2/u1 u3/u2 u_n/u_{n-1} -- ------ ------ ---------- 1 - u2 u3 u_{n} 1 + -- - 1 + -- - ... - 1 + ------- u1 u2 u_{n-1} which we can transform into u1 u2 u1u3 u2u4 u_{n-2}u_{n} -- ------- ------- ------- ---------------, 1 - u1 + u2 - u2 + u3 - u3 + u4 - ... - u_{n-1} + u_{n} a result given by euler. 2. in this case the sum to n terms of the series is equal to the n^{th} convergent of the fraction. there is, however, a different way in which a series may be represented by a continued fraction. we may require to represent the infinite convergent power series a0 + a1x + a2x2 + ... by an infinite continued fraction of the form [beta]0 [beta]1 x [beta]2 x [beta]3 x ------- --------- --------- --------- 1 - 1 - 1 - 1 - ... here the fraction converges to the sum to infinity of the series. its n^{th} convergent is not equal to the sum to n terms of the series. expressions for [beta]0, [beta]1, [beta]2, ... by means of determinants have been given by t. muir (_edinburgh transactions_, vol. xxvii.). a method was given by j. h. lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series. it is practically identical with that of finding the greatest common measure of two polynomials. as an instance leading to results of some importance consider the series x x2 f(n,x) = 1 + --------------- + -------------------------------- + ... ([gamma] + n)1! ([gamma] + n)([gamma] + n + 1)2! we have x f(n + 1,x) - f(n,x) = - ------------------------------ f(n + 2,x), ([gamma] + n)([gamma] + n + 1) whence we obtain f(1,x) 1 x/[gamma]([gamma] + 1) x/([gamma] + 1)([gamma] + 2) ------ = -- ---------------------- ---------------------------- f(0,x) 1 + 1 + 1 + ..., which may also be written [gamma] x x ------- ----------- ----------- [gamma] + [gamma] + 1 + [gamma] + 2 + ... by putting ± x2/4 for x in f(0,x) and f(1,x), and putting at the same time [gamma] = 1/2, we obtain x x2 x2 x2 x x2 x2 x2 tan x = -- -- -- -- tanh x = -- -- -- -- 1 - 3 - 5 - 7 - ... 1 + 3 + 5 + 7 + ... these results were given by lambert, and used by him to prove that [pi] and [pi]2 incommensurable, and also any commensurable power of e. gauss in his famous memoir on the hypergeometric series f([alpha], [beta], [gamma], x) = [alpha]·[beta] [alpha]([alpha] + 1)[beta]([beta] + 1) --------------x + -------------------------------------- x2 + ... 1.[gamma] 1.2.[gamma]·([gamma] + 1) gave the expression for f([alpha], [beta] + 1, [gamma] + 1, x) ÷ f([alpha], [beta], [gamma], x) as a continued fraction, from which if we put [beta] = 0 and write [gamma] - 1 for [gamma], we get the transformation [alpha] [alpha]([alpha] + 1) 1 + -------x + --------------------x2 + [gamma] [gamma]([gamma] + 1) [alpha]([alpha] + 1)([alpha] + 2) ---------------------------------x3 + ... = [gamma]([gamma] + 1)([gamma] + 2) 1 [beta]1 x [beta]2 x -- --------- --------- where 1 - 1 - 1 - ... [alpha] ([alpha] + 1)[gamma] [beta]1 = -------, [beta]3 = --------------------------, ..., [gamma] ([gamma] + 1)([gamma] + 2) ([alpha] + n - 1)([gamma] + n - 2) [beta]_{2n-1} = ------------------------------------, ([gamma] + 2n - 3)([gamma] + 2n - 2) [gamma] - [alpha] 2([gamma] + 1 - [alpha]) [beta]2 = --------------------, [beta]4 = --------------------------, [gamma]([gamma] + 1) ([gamma] + 2)([gamma] + 3) n([gamma] + n - 1 - [alpha]) ..., [beta]_{2n} = ------------------------------------. ([gamma] + 2n - 2)([gamma] + 2n - 1) from this we may express several of the elementary series as continued fractions; thus taking [alpha] = 1, [gamma] = 2, and putting x for -x, x 12x 12x 22x 22x 32x 32x we have log(1 + x) = -- --- --- --- --- --- --- 1 + 2 + 3 + 4 + 5 + 6 + 7 + ... taking [gamma] = 1, writing x/[alpha] for x and increasing [alpha] indefinitely, we have 1 x x x x x e^x = -- -- -- -- -- -- 1 - 1 + 2 - 3 + 2 - 5 + ... for some recent developments in this direction the reader may consult a paper by l. j. rogers in the _proceedings of the london mathematical society_ (series 2, vol. 4). _ascending continued fractions._ there is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction. it is of no interest or importance, though both lambert and lagrange devoted some attention to it. the notation for this type of fraction is b5 + b4 + ---- a5 b3 + --------- a4 b2 + -------------- a3 a1 + ------------------- a2 it is obviously equal to the series b2 b3 b4 b5 a1 + -- + ---- + ------ + -------- + ... a2 a2a3 a2a3a4 a2a3a4a5 _historical note._ the invention of continued fractions is ascribed generally to pietro antonia cataldi, an italian mathematician who died in 1626. he used them to represent square roots, but only for particular numerical examples, and appears to have had no theory on the subject. a previous writer, rafaello bombelli, had used them in his treatise on algebra (about 1579), and it is quite possible that cataldi may have got his ideas from him. his chief advance on bombelli was in his notation. they next appear to have been used by daniel schwenter (1585-1636) in a _geometrica practica_ published in 1618. he uses them for approximations. the theory, however, starts with the publication in 1655 by lord brouncker of the continued fraction 1 12 32 52 -- -- -- -- as an equivalent of [pi]/4. this he is supposed 1 + 2 + 2 + 2 + ... to have deduced, no one knows how, from wallis' formula for 3 . 3 . 5 . 5 . 7 . 7 ... 4/[pi], viz. ------------------------- 2 . 4 . 4 . 6 . 6 . 8 ... john wallis, discussing this fraction in his _arithmetica infinitorum_ (1656), gives many of the elementary properties of the convergents to the general continued fraction, including the rule for their formation. huygens (_descriptio automati planetarii_, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his _planetarium_. nicol saunderson (1682-1739), euler and lambert helped in developing the theory, and much was done by lagrange in his additions to the french edition of euler's _algebra_ (1795). moritz a. stern wrote at length on the subject in _crelle's journal_ (x., 1833; xi., 1834; xviii., 1838). the theory of the convergence of continued fractions is due to oscar schlomilch, p. f. arndt, p. l. seidel and stern. o. stolz, a. pringsheim and e. b. van vleck have written on the convergence of infinite continued fractions with complex elements. references.--for the further history of continued fractions we may refer the reader to two papers by gunther and a. n. favaro, _bulletins di bibliographia e di storia delle scienze mathematische e fisicke_, t. vii., and to m. cantor, _geschichte der mathematik_, 2nd bd. for text-books treating the subject in great detail there are those of g. chrystal in english; serret's _cours d`algebre superieure_ in french; and in german those of stern, schlomilch, hatterdorff and stolz. for the application of continued fractions to the theory of irrational numbers there is p. bachmann's _vorlesungen uber die natur der irrationalzahnen_ (1892). for the application of continued fractions to the theory of lenses, see r. s. heath's _geometrical optics_, chaps. iv. and v. for an exhaustive summary of all that has been written on the subject the reader may consult bd. 1 of the _encyklopadie der mathematischen wissenschaften_ (leipzig). (a. e. j.)