INTERPLEADER

interpleader, in english law, the form of action by which a person who is sued at law by two or more parties claiming adversely to each other for the recovery of money or goods wherein he has no interest, obtains relief by procuring the rival claimants to try their rights between or among themselves only. originally the only relief available to the possessor against such adverse claims was by means of a bill of interpleader in equity. the interpleader act 1831 enabled the defendant in such cases, on application to the court, to have the original action stayed and converted into a trial between the two claimants. the common law procedure act of 1860 further extended the power of the common law courts in interpleader; and the judicature act 1875 enacted that the practice and procedure under these two statutes should apply to all divisions of the high court of justice. the judicature act also extended the remedy of interpleader to a debtor or other person liable in respect of a debt alleged to be assigned, when the assignment was disputed. in 1883 the acts of 1831 and 1860 were embodied in the form of rules by the _rules of the supreme courts_ (1883), o. lvii. by reference to which all questions of interpleader in the high court of justice are now determined. the acts themselves were repealed by the statute law revision act of the same year. interpleader is the equivalent of multiplepoinding in scots law. interpolation (from lat. _interpolare_, to alter, or insert something fresh, connected with _polire_, a polish), in mathematics, the process of obtaining intermediate terms of a series of which particular terms only are given. the cubes, for instance, shown in the second column of the accompanying table, may +-------+----------------+ |number.| cube of number.| +-------+----------------+ | 0 | 0 | | 1 | 1 | | 2 | 8 | | 3 | 27 | | 4 | 64 | | 5 | 125 | | 6 | 216 | | . | . | | . | . | | . | . | +-------+----------------+ be regarded as terms of a series, and the cube of a fractional number, not exceeding the last number in the first column, may be found by interpolation. the process of obtaining the cube of a number exceeding the last number in the first column would be _extrapolation_; the formulae which apply to interpolation apply in theory to extrapolation, but in practice special precautions as to accuracy are necessary. the present article deals only with interpolation. the term is usually limited to those cases in which there are two quantities, x and u, which are so related that when x has any arbitrary value, lying perhaps between certain limits, the value of u is determinate. there is a given series of associated values of u and of x, and interpolation consists in determining the value of u for any arbitrary value of x, or the value of x for any arbitrary value of u, lying between two of the values in the series. either of the two quantities may be regarded as a function of the other; it is convenient to treat one, x, as the "independent variable," the other, u, being treated as the "dependent variable," i.e. as a function of x. if, as is usually the case, the successive values of one of the quantities proceed by a constant increment, this quantity is to be regarded as the independent variable. the two series of values may be tabulated, those of x being placed in a column (or row), and those of u in a parallel column (or row); u is then said to be _tabulated in terms of_ x. the independent variable x is called the _argument_, and the dependent variable u is called the _entry_. interpolation, in the ordinary sense, consists in determining the value of u for a value of x intermediate between two values appearing in the table. this may be described as _direct interpolation_, to distinguish it from _inverse interpolation_, which consists in determining the value of x for a value of u intermediate between two in the table. the methods employed can be extended to cases in which the value of u depends on the values of two or more independent quantities x, y,... in the ordinary case we may regard the values of x as measured along a straight line ox from a fixed point o, so that to any value of x there corresponds a point on the line. if we represent the corresponding value of u by an ordinate drawn from the line, the extremities of all such ordinates will lie on a curve which will be the graph of u with regard to x. interpolation therefore consists in determining the length of the ordinate of a curve occupying a particular position, when the lengths of ordinates occupying certain specified positions are known. if u is a function of two variables, x and y, we may similarly represent it by the ordinate of a surface, the position of the ordinate being determined by the values of x and of y jointly. the series or tables to which interpolation has to be applied may for convenience be regarded as falling into two main groups. the first group comprises mathematical tables, i.e. tables of mathematical functions; in the case of such a table the value of the function u for each tabulated value of x is calculated to a known degree of accuracy, and the degree of accuracy of an interpolated value of u can be estimated. the second group comprises tables of values which are found experimentally, e.g. values of a physical quantity or of a statistical ratio; these values are usually subject to certain "errors" of observation or of random selection (see probability). the methods of interpolation are usually the same in the two groups of cases, but special considerations have to be taken into account in the second group. the line of demarcation of the two groups is not absolutely fixed; the tables used by actuaries, for instance, which are of great importance in practical life, are based on statistical observations, but the tables formed directly from the observations have been "smoothed" so as to obtain series which correspond in form to the series of values of mathematical functions. it must be assumed, at any rate in the case of a mathematical function, that the "entry" u varies continuously with the "argument" x, i.e. that there are no sudden breaks, changes of direction, &c., in the curve which is the graph of u. various methods of interpolation are described below. the simplest is that which uses the _principle of proportional parts_; and mathematical tables are usually arranged so as to enable this method to be employed. where this is not possible, the methods are based either on the use of taylor's theorem, which gives a formula involving differential coefficients (see infinitesimal calculus), or on the properties of finite differences (see differences, calculus of). taylor's theorem can only be applied directly to a known mathematical function; but it can be applied indirectly, by means of finite differences, in various cases where the form of the function expressing u in terms of x is unknown; and even where the form of this function is known it is sometimes more convenient to determine the differential coefficients by means of the differences than to calculate them directly from their mathematical expressions. finally, there are cases where we cannot even employ finite-difference formulae directly. in these cases we must adopt some special method; e.g. we may instead of u tabulate some function of u, such as its logarithm, which is found to be amenable to ordinary processes, then determine the value of this function corresponding to the particular value of x, and thence determine the corresponding value of u itself. in considering methods of interpolation, it will be assumed, unless the contrary is stated, that the values of x proceed by a constant increment, which will be denoted by h. in order to see what method is to be employed, it is usually necessary to arrange the given series of values of u in the form of a table, as explained above, and then to take the successive _differences_ of u. the differences of the successive values of u are called its _first differences_; these form a new series, the first differences of which are the _second differences_ of u; and so on. the systems of notation of the differences are explained briefly below. for the fuller discussion, reference should be made to differences, calculus of. i. interpolation from mathematical tables a. direct interpolation. 1. _interpolation by first differences._--the simplest cases are those in which the first difference in u is constant, or nearly so. for example:-- _example_ 1.--(u = log_10 x). _example_ 2.--(u = log_10 x). +-------+--------+---------+ +------+--------+---------+ | x. | u. |1st diff.| | x. | u. |1st diff.| +-------+--------+---------+ +------+--------+---------+ | | | + | | | | + | | 4.341 |.6375898| | | 7.40 | .86923 | | | | | 1000 | | | | 59 | | 4.342 |.6376898| | | 7.41 | .86982 | | | | | 1000 | | | | 58 | | 4.343 |.6377898| | | 7.42 | .87040 | | | | | 1000 | | | | 59 | | 4.344 |.6378898| | | 7.43 | .87099 | | | | | 1000 | | | | 58 | | 4.345 |.6379898| | | 7.44 | .87157 | | +-------+--------+---------+ +------+--------+---------+ in example 1 the first difference of u corresponding to a difference of h [equivalent] .001 in x is .0001000; but, since we are working throughout to seven places of decimals, it is more convenient to write it 1000. this system of ignoring the decimal point in dealing with differences will be adopted throughout this article. to find u for an intermediate value of x we assume the principle of proportional parts, i.e. we assume that the difference in u is proportional to the difference in x. thus for x = 4.342945 the difference in u is .945 of 1000 = 945, so that u is .6376898 + .0000945 = .6377843. for x = 4.34294482 the difference in u would be 944.82, so that the value of u would apparently be .6376898 + .000094482 = .637784282. this, however, would be incorrect. it must be remembered that the values of u are only given "correct to seven places of decimals," i.e. each tabulated value differs from the corresponding true value by a _tabular error_ which may have any value up to [+-] 1/2 of .0000001; and we cannot therefore by interpolation obtain a result which is correct to nine places. if the interpolated value of u has to be used in calculations for which it is important that this value should be as accurate as possible, it may be convenient to retain it temporarily in the form .6376898 + 944 82 = .6377842 82 or .6376898 + 944^82 = .6377842^82; but we must ultimately return to the seven-place arrangement and write it as .6377843. the result of interpolation by first difference is thus usually subject to two inaccuracies, the first being the tabular error of u itself, and the second being due to the necessity of adjusting the final figure of the added (proportional) difference. if the tabulated values are correct to seven places of decimals, the interpolated value, with the final figure adjusted, will be within .0000001 of its true value. in example 2 the differences do not at first sight appear to run regularly, but this is only due to the fact that the final figure in each value of u represents, as explained in the last paragraph, an approximation to the true value. the general principle on which we proceed is the same; but we use the actual difference corresponding to the interval in which the value of x lies. thus for x = 7.41373 we should have u = .86982 + (.373 of 58) = .87004; this result being correct within .00001. 2. _interpolation by second differences._--if the consecutive first differences of u are not approximately equal, we must take account of the next order of differences. for example:-- _example 3._--(u = log_10 x). +-----+-------+---------+---------+ | x. | u. |1st diff.|2nd diff.| +-----+-------+---------+---------+ | 6.0 |.77815 | | | | | | +718 | | | 6.1 |.78533 | | -12 | | | | +706 | | | 6.2 |.79239 | | -11 | | | | +695 | | | 6.3 |.79934 | | -11 | | | | +684 | | | 6.4 |.80618 | | -11 | | | +673 | | | 6.5 |.81291 | | | +-----+-------+---------+---------+ in such a case the _advancing-difference_ formula is generally used. the notation is as follows. the series of values of x and of u are respectively x0, x1, x2, ... and u0, u1, u2, ... ; and the successive differences of u are denoted by [delta]u, [delta]^2u, ... thus [delta]u0 denotes u1 - u0, and [delta]^2u0 denotes [delta]u1 - [delta]u0 = u2 - 2u1 + u0. the value of x for which u is sought is supposed to lie between x0 and x1. if we write it equal to x0 + [theta](x1 - x0) = x0 + [theta]h, so that [theta] lies between 0 and 1, we may denote it by x_([theta]), and the corresponding value of u by u_([theta]). we have then [theta] (1 - [theta]) u[theta] = u0 + [theta][delta]u0 - --------------------- [delta]^2u0 2! [theta] (1 - [theta]) (2 - [theta]) + ----------------------------------- [delta]^3u0 - ... (1). 3! tables of the values of the coefficients of [delta]^2u0 and [delta]^3u0 to three places of decimals for various values of [theta] from 0 to 1 are given in the ordinary collections of mathematical tables; but the formula is not really convenient if we have to go beyond [delta]^2u0, or if [delta]^2u0 itself contains more than two significant figures. to apply the formula to example 3 for x = 6.277, we have [theta] = .77, so that u_([theta]) = .79239 + (.77 of 695) - (.089 of -11) = .79239 + 535 15 + 0 98 = .79775. here, as elsewhere, we use two extra figures in the intermediate calculations, for the purpose of adjusting the final figure in the ultimate result. 3. _taylor's theorem._--where differences beyond the second are involved, taylor's theorem is useful. this theorem (see infinitesimal calculus) gives the formula [theta]^2 [theta]^3 u_([theta]) = u0 + c1[theta] + c2 --------- + c3 --------- + ... (2), 2! 3! where, c1, c2, c3, ... are the values for x = x0 of the first, second, third, ... differential coefficients of u with regard to x. the values of c1, c2, ... can occasionally be calculated from the analytical expressions for the differential coefficients of u; but more generally they have to be calculated from the tabulated differences. for this purpose _central-difference_ formulae are the best. if we write [mu][delta]u0 = (1/2)([delta]u0 + [delta]u_(-1)) \ [delta]^2u0 = [delta]^2u_(-1) | (3), [mu][delta]^3u0 = (1/2)([delta]^3u_(-1) + [delta]^3u_(-2)) | &c. / so that, if (as in ss 1 and 2) each difference is placed opposite the space between the two quantities of which it is the difference, the expressions [delta]^2u0, [delta]^4u0, ... denote the differences of even order in a horizontal line with u0, and [mu][delta]u0, [mu][delta]^3u0, ... denote the means of the differences of odd order immediately below and above this line, then (see differences, calculus of) the values of c1, c2, . . . are given by c1 = [mu][delta]u0 - (1/6)[mu][delta]^3u0 + (1/30)[mu][delta]^5u0 \ - (1/140)[mu][delta]^7u0 + ... | c2 = [delta]^2u0 - (1/12)[delta]^4u0 + (1/90)[delta]^6u0 - (1/560)[delta]^8u0 + ... | c3 = [mu][delta]^3u0 - (1/4)[mu][delta]^5u0 + (7/120)[mu][delta]^7u0 - ... | c4 = [delta]^4u0 - (1/6)[delta]^6u0 + (7/240)[delta]^8u0 - ... | c5 = [mu][delta]^5u0 - (1/3)[mu][delta]^7u0 + ... | (4). c6 = [delta]^6u0 - (1/4)[delta]^8u0 + ... | . . | . . | . . / if a calculating machine is used, the formula (2) is most conveniently written u_([theta]) = u0 + p1[theta] \ p1 = c1 + (1/2)p2[theta] | p2 = c2 + (1/3)p3[theta] | (5). . . | . . | . . / using [theta] as the multiplicand in each case, the successive expressions ... p3, p2, p1, u_([theta]) are easily calculated. as an example, take u = tan x to five places of decimals, the values of x proceeding by a difference of 1 deg. it will be found that the following is part of the table:-- _example_ 4.--(u = tan x). +--------+--------+---------+---------+---------+---------+ | x. | u. |1st diff.|2nd diff.|3rd diff.|4th diff.| +--------+--------+---------+---------+---------+---------+ | | | + | + | + | + | | 65 deg.| 2.14451| | 732 | | 16 | | | | 10153 | | 96 | | | 66 deg.| 2.24604| | 828 | | 19 | | | | 10981 | | 115 | | | 67 deg.| 2.35585| | 943 | | 18 | +--------+--------+---------+---------+---------+---------+ to find u for x = 66 deg. 23', we have [theta] = 23/60 = .3833333. the following shows the full working: in actual practice it would be abbreviated. the operations commence on the right-hand side. it will be noticed that two extra figures are retained throughout. u0. [mu][delta]u0. [delta]^2u0. [mu][delta]^3u0. [delta]^4u0. 2.24604 +10567^00 +828^00 +105^50 +19^00 - 17^58 - 1^58 --------- ------- --------- ------ c1 = +10549^42 c2 = +826^42 c3 = +105^50 c4 = +19^00 p1[theta] = +4105^67 (1/2)p2[theta] = + 161^02 (1/3)p3[theta] = + 13^71 (1/8)c4[theta] = + 1^82 -------- --------- ------- ------- u_[theta] = 2.28710 p1 = +10710^44 p2 = +840^13 p3 = +107^32 the value 2.2870967, obtained by retaining the extra figures, is correct within .7 of .00001 (s 8), so that 2.28710 is correct within .00001 1. in applying this method to mathematical tables, it is desirable, on account of the tabular error, that the differences taken into account in (4) should end with a difference of even order. if, e.g. we use [mu][delta]^3u0 in calculating c1 and c3, we ought also to use [delta]^4u0 for calculating c2 and c4, even though the term due to [delta]^4u0 would be negligible if [delta]^4u0 were known exactly. 4. _geometrical and algebraical interpretation._--in applying the principle of proportional parts, in such a case as that of example 1, we in effect treat the graph of u as a straight line. we see that the extremities of a number of consecutive ordinates lie approximately in a straight line: i.e. that, if the values are correct within [+-](1/2)[rho], a straight line passes through points which are within a corresponding distance of the actual extremities of the ordinates; and we assume that this is true for intermediate ordinates. algebraically we treat u as being of the form a + bx, where a and b are constants determined by the values of u at the extremities of the interval through which we interpolate. in using first and second differences we treat u as being of the form a + bx + cx^2; i.e. we pass a parabola (with axis vertical) through the extremities of three consecutive ordinates, and consider that this is the graph of u, to the degree of accuracy given by the data. similarly in using differences of a higher order we replace the graph by a curve whose equation is of the form u = a + bx + cx^2 + dx^3 + ... the various forms that interpolation-formulae take are due to the various principles on which ordinates are selected for determining the values of a, b, c ... b. _inverse interpolation._ 5. to find the value of x when u is given, i.e. to find the value of [theta] when u_([theta]) is given, we use the same formula as for direct interpolation, but proceed (if differences beyond the first are involved) by successive approximation. taylor's theorem, for instance, gives [theta] [theta] = (u_[theta] - u0) / (c1 + c2 + ------- + ...) 2! = (u_[theta] - u0) / p1 (6), we first find an approximate value for [theta]: then calculate p1, and find by (6) a more accurate value of [theta]; then, if necessary, recalculate p1, and thence [theta], and so on. ii. construction of tables by subdivision of intervals 6. when the values of u have been tabulated for values of x proceeding by a difference h, it is often desirable to deduce a table in which the differences of x are h/n, where n is an integer. if n is even it may be advisable to form an intermediate table in which the intervals are (1/2)h. for this purpose we have u_(1/2) = (1/2)(u0 + u1) (7), where u = u - (1/8)[delta]^2u + (3/128)[delta]^4u - (5/1024)[delta]^6u + ... = u - (1/8)[[delta]^2u - (3/16){[delta]^4u - (5/24)([delta]^6u - ...)}] (8). the following is an example; the data are the values of tan x to five places of decimals, the interval in x being 1 deg. the differences of odd order are omitted for convenience of printing. _example 5._ +--------+---------+-----------+-----------+-----------+------------+------------+-------------+ | | u [eq] | | | | |u = mean of | | | x. | tan x. |[delta]^2u.|[delta]^4u.|[delta]^6u.| u. |values of u.| x. | +--------+---------+-----------+-----------+-----------+------------+------------+-------------+ | | | + | + | + | | | | | 73 deg.| 3.27085 | 2339 | 100 | 5 | 3.26794 95 | | | | | | | | | | 3.37594 | 73(1/2) deg.| | 74 deg.| 3.48741 | 2808 | 132 | 23 | 3.48392 98 | | | | | | | | | | 3.60588 | 74(1/2) deg.| | 75 deg.| 3.73205 | 3409 | 187 | 18 | 3.72783 17 | | | | | | | | | | 3.86671 | 75(1/2) deg.| | 76 deg.| 4.01078 | 4197 | 260 | 51 | 4.00559 22 | | | | | | | | | | 4.16530 | 76(1/2) deg.| | 77 deg.| 4.33148 | 5245 | 384 | 64 | 4.32501 07 | | | +--------+---------+-----------+-----------+-----------+------------+------------+-------------+ if a new table is formed from these values, the intervals being 1/2 deg., it will be found that differences beyond the fourth are negligible. to subdivide h into smaller intervals than (1/2)h, various methods may be used. one is to calculate the sets of quantities which in the new table will be the successive differences, corresponding to u0, u1, ... and to find the intermediate terms by successive additions. a better method is to use a formula due to j. d. everett. if we write [phi] = 1 - [theta], everett's formula is, in its most symmetrical form, ([theta] + 1)[theta]([theta] - 1) u_([theta]) = [theta]u1 + ---------------------------------[delta]^2u1 3! ([theta] + 2)([theta] + 1)[theta]([theta] - 1)([theta] - 2) + -----------------------------------------------------------[delta]^4u1 + ... ([phi] + 1)[phi]([phi] - 1) + [phi]u0 + ---------------------------[delta]^2u0 3! ([phi] + 2)([phi] + 1)[phi]([phi] - 1)([phi] - 2) + -------------------------------------------------[delta]^4u0 + ... (9). 5! for actual calculations a less symmetrical form may be used. denoting ([theta] + 1)[theta]([theta] - 1) ---------------------------------[delta]^2u1 3! ([theta] + 2)([theta] + 1)[theta]([theta] - 1)([theta] - 2) + -----------------------------------------------------------[delta]^4u1 + ... (10) 5! by _([theta])v1, we have, for interpolation between u0 and u1, u_([theta]) = u0 + [theta][delta]u0 + _([theta])v1 + _(1 - [theta])v0 (11), the successive values of [theta] being 1/n, 2/n, ... (n-1)/n. for interpolation between u1 and u2 we have, with the same succession of values of [theta], u_(1+[theta]) = u1 + _([theta])v1, v2 + _(1-[theta])v1 (12). the values of _(1-[theta])v1 in (12) are exactly the same as those of ([theta])v1 in (11), but in the reverse order. the process is therefore that (i.) we find the successive values of u0 + [theta][delta]u0, &c., i.e. we construct a table, with the required intervals of x, as if we had only to take first differences into account; (ii.) we construct, in a parallel column, a table giving the values of _([theta])v1, &c.; (iii.) we repeat these latter values, placing the set belonging to each interval h in the interval next following it, and writing the values in the reverse order; and (iv.) by adding horizontally we get the final values for the new table. as an example, take the values of tan x by intervals of 1/2 deg. in x, as found above (ex. 5). the first diagram below is a portion of this table, with the differences, and the second shows the calculation of the terms of (11) so as to get a table in which the intervals are 0.1 of 1 deg. the last column but one in the second diagram is introduced for convenience of calculation. _example 6._ +---------+-----------+---------+-----------+-----------+-----------+ | x. | u = tan x.|[delta]u.|[delta]^2u.|[delta]^3u.|[delta]^4u.| +---------+-----------+---------+-----------+-----------+-----------+ | | | + | + | + | + | | | | 11147 | | 62 | | | 74 deg.0| 3.48741 | | 700 | | 8 | | | | 11847 | | 70 | | | 74 deg.5| 3.60588 | | 770 | | 9 | | | | 12617 | | 79 | | +---------+-----------+---------+-----------+-----------+-----------+ +----------+------------------+--------------+----------------+----------------+---------+ | | u0 + | | | _([theta])v1 + | | | x. | [theta][delta]u0.| _([theta])v1.| _(1-[theta])v0.| _(1-[theta])v0.| u. | +----------+------------------+--------------+----------------+----------------+---------+ | 73 deg.6 | . | -22 35 | . | . | . | | 73 deg.7 | . | -39 11 | . | . | . | | 73 deg.8 | . | -44 71 | . | . | . | | 73 deg.9 | . | -33 54 | . | . | . | | 74 deg.0 | 3.48741 00 | | | | 3.48741 | | 74 deg.1 | 3.51110 40 | -24 58 | -33 54 | -58 12 | 3.51052 | | 74 deg.2 | 3.53479 80 | -43 02 | -44 71 | -87 73 | 3.53392 | | 74 deg.3 | 3.55849 20 | -49 18 | -39 11 | -88 29 | 3.55761 | | 74 deg.4 | 3.58218 60 | -36 89 | -22 35 | -59 24 | 3.58159 | | 74 deg.5 | 3.60588 00 | | | | 3.60588 | +----------+------------------+--------------+----------------+----------------+---------+ the following are the values of the coefficients of u1, [delta]^2u1, [delta]^4u1, and [delta]^6u1 in (9) for certain values of n. for calculating the four terms due to [delta]^2u1 in the case of n = 5 it should be noticed that the third term is twice the first, the fourth is the mean of the first and the third, and the second is the mean of the third and the fourth. in table 3, and in the last column of table 2, the coefficients are corrected in the last figure. table 1.--n = 5. +------+---------------+---------------+--------------------------+ |co. u.|co. [delta]^2u.|co. [delta]^4u.| co. [delta]^6u. | +------+---------------+---------------+--------------------------+ | + | - | + | - | | .2 | .032 | .006336 | .00135168 = 1/740 approx.| | .4 | .056 | .010752 | .00226304 = 1/442 " | | .6 | .064 | .011648 | .00239616 = 1/417 " | | .8 | .048 | .008064 | .00160512 = 1/623 " | +------+---------------+---------------+--------------------------+ table 2.--n = 10. +------+---------------+---------------+---------------+ |co. u.|co. [delta]^2u.|co. [delta]^4u.|co. [delta]^6u.| +------+---------------+---------------+---------------+ | + | - | + | - | | .1 | .0165 | .00329175 | .000704591 | | .2 | .0320 | .00633600 | .001351680 | | .3 | .0455 | .00889525 | .001887064 | | .4 | .0560 | .01075200 | .002263040 | | .5 | .0625 | .01171875 | .002441406 | | .6 | .0640 | .01164800 | .002396160 | | .7 | .0595 | .01044225 | .002115799 | | .8 | .0480 | .00806400 | .001605120 | | .9 | .0285 | .00454575 | .000886421 | +------+---------------+---------------+---------------+ table 3.--n = 12. +------+---------------+---------------+---------------+ |co. u.|co. [delta]^2u.|co. [delta]^4u.|co. [delta]^6u.| +------+---------------+---------------+---------------+ | + | - | + | - | | 1/12 | .013792438 | .002753699 | .000589623 | | 2/12 | .027006173 | .005363726 | .001145822 | | 3/12 | .039062500 | .007690430 | .001636505 | | 4/12 | .049382716 | .009602195 | .002032211 | | 5/12 | .057388117 | .010979463 | .002307357 | | 6/12 | .062500000 | .011718750 | .002441406 | | 7/12 | .064139660 | .011736667 | .002419911 | | 8/12 | .061728395 | .010973937 | .002235432 | | 9/12 | .054687500 | .009399414 | .001888275 | |10/12 | .042438272 | .007014103 | .001387048 | |11/12 | .024402006 | .003855178 | .000748981 | +------+---------------+---------------+---------------+ iii. general observations 7. _derivation of formulae._--the advancing-difference formula (1) may be written, in the symbolical notation of finite differences, u_[theta] = (1 + [delta])^([theta])u0 = e^([theta])u0 (13); and it is an extension of the theorem that if n is a positive integer n(n - 1) u_n = u0 + n[delta]u0 + --------[delta]^2u0 + ... (14), 2! the series being continued until the terms vanish. the formula (14) is identically true: the formula (13) or (1) is only formally true, but its applicability to concrete cases is due to the fact that the series in (1), when taken for a definite number of terms, differs from the true value of u_([theta]) by a "remainder" which in most cases is very small when this definite number of terms is properly chosen. everett's formula (9), and the central-difference formula obtained by substituting from (4) in (2), are modifications of a standard formula [theta]([theta] - 1) u_[theta] = u0 + [theta][delta]u_(1/2) + --------------------[delta]^2u0 + 2! ([theta] + 1)[theta]([theta] - 1) ----------------------------------[delta]^3u_(1/2) + 3! ([theta] + 1)[theta]([theta] - 1)([theta] - 2) ----------------------------------------------[delta]^4u0 + ... (15) 4! which may similarly be regarded as an extension of the theorem that, if n is a positive integer, n(n - 1) (n + 1)n(n - 1) u_n = u0 + n[delta]u_(1/2) + --------[delta]^2u0 + ---------------[delta]^3u_(1/2) + ... (16). 2! 3! there are other central-difference formulae besides those mentioned above; the general symbolical expression is u_[theta] = (cosh [theta]hd + sinh [theta]hd)u0 (17), where cosh (1/2)hd = [mu], sinh (1/2)hd = (1/2)[delta] (18). 8. _comparative accuracy._--central-difference formulae are usually more accurate than advancing-difference formulae, whether we consider the inaccuracy due to omission of the "remainder" mentioned in the last paragraph or the error due to the approximative character of the tabulated values. the latter is the more important. if each tabulated value of u is within [+-](1/2)[rho] of the corresponding true value, and if the differences used in the formulae are the tabular differences, i.e. the actual successive differences of the tabulated values of u, then the ratio of the limit of error of u_([theta]), as calculated from the first r terms of the series in (1), to (1/2)[rho] is the sum of the first r terms of the series 1 + o + [theta](1 - [theta]) + [theta](1 - [theta])(2 - [theta]) + (7/12)[theta](1 - [theta])(2 - [theta])(3 - [theta]) + (1/4)[theta](1 - [theta])(2 - [theta])(3 - [theta])(4 - [theta]) + (31/360)[theta](1 - [theta])...(5 - [theta])+ ..., while the corresponding ratio for the use of differences up to [delta]^(2p)u0 inclusive in (4) or up to [delta]^(2p)u1 and o^(2p)u0 in (9) (i.e. in effect, up to [delta]^(2p + 1)u(1/2)) is the sum of the first p + 1 terms of the series [theta](1 - [theta]) (1 + [theta])[theta](1 - [theta])(2 -[theta]) 1 + -------------------- + --------------------------------------------- + 1.1 (2!)^2 (2 + [theta])(1 + [theta])[theta](1 - [theta])(2 - [theta])(3 -[theta]) ----------------------------------------------------------------------- + ..., (3!)^2 it being supposed in each case that [theta] lies between 0 and 1. the following table gives a comparison of the respective limits of error; the lines i. and ii. give the errors due to the advancing-difference and the central-difference formulae, and the coefficient [rho] is omitted throughout. table 4. +----------+------------------------------------------------+ | | error due to use of differences up to and | | | including | | +------+------+------+------+------+------+------+ | | 1st. | 2nd. | 3rd. | 4th. | 5th. | 6th. | 7th. | +----------+------+------+------+------+------+------+------+ |.5 / i. | .500 | .625 | .813 |1.086 |1.497 |2.132 |3.147 | | \ ii. | .500 | .625 | .625 | .696 | .696 | .745 | .745 | |.2 / i. | .500 | .580 | .724 | .960 |1.343 |1.976 |3.042 | | \ ii. | .500 | .580 | .580 | .624 | .624 | .653 | .653 | |.4 / i. | .500 | .620 | .812 |1.104 |1.553 |2.265 |3.422 | | \ ii. | .500 | .620 | .620 | .688 | .688 | .734 | .734 | |.6 / i. | .500 | .620 | .788 |1.024 |1.366 |1.886 |2.700 | | \ ii. | .500 | .620 | .620 | .688 | .688 | .734 | .734 | |.8 / i. | .500 | .580 | .676 | .800 | .969 |1.213 |1.582 | | \ ii. | .500 | .580 | .580 | .624 | .624 | .653 | .653 | +----------+------+------+------+------+------+------+------+ in some cases the differences tabulated are not the tabular differences, but the corrected differences; i.e. each difference, like each value of u, is correct within [+-](1/2)[rho]. it does not follow that these differences should be used for interpolation. whatever formula is employed, the first difference should always be the tabular first difference, not the corrected first difference; and, further, if a central-difference formula is used, each difference of odd order should be the tabular difference of the corrected differences of the next lower order. (this last result is indirectly achieved if everett's formula is used.) with these precautions (i.) the central-difference formula is slightly improved by using corrected instead of tabular differences, and (ii.) the advancing-difference formula is greatly improved, being better than the central-difference formula with tabular differences, but still not so good as the latter with corrected differences. for [theta] = .5, for instance, supposing we have to go to fifth differences, the limits [+-]1.497 and [+-].696, as given above, become [+-].627 and [+-].575 respectively. 9. _completion of table of differences._--if no values of u outside the range within which we have to interpolate are given, the series of differences will be incomplete at both ends. it may be continued in each direction by treating as constant the extreme difference of the highest order involved; and central-difference formulae can then be employed uniformly throughout the whole range. suppose, for instance, that the values of tan x in s 6 extended only from x = 60 deg. to x = 80 deg., we could then complete the table of differences by making the entries shown in italics below. _example 7._ +--------+---------+---------+-----------+-----------+-----------+-----------+-----------+ | x. | tan x. |[delta]u.|[delta]^2u.|[delta]^3u.|[delta]^4u.|[delta]^5u.|[delta]^6u.| +--------+---------+---------+-----------+-----------+-----------+-----------+-----------+ | | | + | + | + | + | + | + | | | | _6775_ | | _34_ | | | | | 60 deg.| 1.73205 | | _425_ | | _9_ | | | | | | 7200 | | _43_ | | | | | 61 deg.| 1.80405 | | 468 | | _9_ | | | | | | 7668 | | 52 | | | | | 62 deg.| 1.88073 | | 520 | | 9 | | | | | | 8188 | | 61 | | | | | 63 deg.| 1.96261 | | 581 | | 10 | | | | | | 8769 | | 71 | | | | | 64 deg.| 2.05030 | . | 652 | . | 9 | | | | . | . | . | . | . | . | . | . | | . | . | . | . | . | . | . | . | | . | . | . | . | . | . | . | . | | 75 deg.| 3.73205 | . | 3409 | . | 187 | . | 18 | | | | 27873 | | 788 | | 73 | | | 76 deg.| 4.01078 | | 4197 | | 260 | | 51 | | | | 32070 | | 1048 | | 124 | | | 77 deg.| 4.33148 | | 5245 | | 384 | | 64 | | | | 37315 | | 1432 | | 188 | | | 78 deg.| 4.70463 | | 6677 | | 572 | | _64_ | | | | 43992 | | 2004 | | _252_ | | | 79 deg.| 5.14455 | | 8681 | | _824_ | | _64_ | | | | 52673 | | _2828_ | | _316_ | | | 80 deg.| 5.67128 | | _11509_ | | _1140_ | | _64_ | | | | _64182_ | | _3968_ | | _380_ | | +--------+---------+---------+-----------+-----------+-----------+-----------+-----------+ for interpolating between x = 60 deg. and x = 61 deg. we should obtain the same result by applying everett's formula to this table as by using the advancing-difference formula; and similarly at the other end for the receding differences. _interpolation by substituted tabulation._ 10. the relation of u to x may be such that the successive differences of u increase rapidly, so that interpolation-formulae cannot be employed directly. other methods have then to be used. the best method is to replace u by some expression v which is a function of u such that (i.) the value of v or of u can be determined for any given value of u or of v, and (ii.) when v is tabulated in terms of x the differences decrease rapidly. we can then calculate v, and thence u, for any intermediate value of x. if, for instance, we require tan x for a value of x which is nearly 90 deg., it will be found that the table of tangents is not suitable for interpolation. we can, however, convert it into a table of cotangents to about the same number of significant figures; from this we can easily calculate cot x, and thence tan x. 11. this method is specially suitable for statistical data, where the successive values of u represent the area of a figure of frequency up to successive ordinates. we have first to determine, by inspection, a curve which bears a general similarity to the unknown curve of frequency, and whose area and abscissa are so related that either can be readily calculated when the other is known. this may be called the _auxiliary curve_. denoting by [xi] the abscissa of this curve which corresponds to area u, we find the value of [xi] corresponding to each of the given values of u. then, tabulating [xi] in terms of x, we have a table in which, if the auxiliary curve has been well chosen, differences of [xi] after the first or second are negligible. we can therefore find [xi], and thence u, for any intermediate value of x. _extensions._ 12. _construction of formulae._--any difference of u of the rth order involves r + 1 consecutive values of u, and it might be expressed by the suffixes which indicate these values. thus we might write the table of differences +----+----+---------+----------+--------------+-----------------+ | x. | u. |1st diff.| 2nd diff.| 3rd diff. | 4th diff. | +----+----+---------+----------+--------------+-----------------+ | . | . | . | . | . | . | | . | . | . | . | . | . | | . | . | . | . | . | . | | . | . | (-1, 0) | . |(-2, -1, 0, 1)| . | | x0 | u0 | |(-1, 0, 1)| |(-2, -1, 0, 1, 2)| | | | (0, 1) | | (-1, 0, 1, 2)| | | x1 | u1 | | (0, 1, 2)| | (-1, 0, 1, 2, 3)| | | | (1, 2) | | (0, 1, 2, 3)| | | x2 | u2 | | (1, 2, 3)| | (0, 1, 2, 3, 4)| | . | . | (2, 3) | . | (1, 2, 3, 4)| . | | . | . | . | . | . | . | | . | . | . | . | . | . | | . | . | . | . | . | . | +----+----+---------+----------+--------------+-----------------+ the formulae (1) and (15) might then be written x - x0 x - x0 x - x1 u = u0 + ------(0, 1) + ------ . ------(0, 1, 2) + h h 2h x - x0 x - x1 x - x2 ------ . ------ . ------(0, 1, 2, 3) + ... (19), h 2h 3h x - x0 x - x0 x - x1 u = u0 + ------(0, 1) + ------ . ------(-1, 0, 1) + h h 2h x - x0 x - x1 x - x_(-1) ------ . ------ . ----------(-1, 0, 1, 2) + ... (20). h 2h 3h the general principle on which these formulae are constructed, and which may be used to construct other formulae, is that (i.) we start with any tabulated value of u, (ii.) we pass to the successive differences by steps, each of which may be either downwards or upwards, and (iii.) the new suffix which is introduced at each step determines the new factor (involving x) for use in the next term. for any particular value of x, however, all formulae which end with the same difference of the _r_th order give the same result, provided tabular differences are used. if, for instance, we go only to first differences, we have x - x0 x - x1 u0 + ------(0, 1) = u1 + ------(0, 1) h h identically. 13. _ordinates not equidistant._--when the successive ordinates in the graph of u are not equidistant, i.e. when the differences of successive values of x are not equal, the above principle still applies, provided the differences are adjusted in a particular way. let the values of x for which u is tabulated be a = x0 + [alpha]h, b = x0 + [beta]h, c = x0 + [gamma]h,... then the table becomes +---------------+----------+---------------------------------------------------+ | | | adjusted differences | | x. | u. +-----------------+--------------------------+------+ | | | 1st diff. | 2nd diff. | &c. | +---------------+----------+-----------------+--------------------------+------+ | . | . | . | . | | | . | . | . | . | | | . | . | . | . | | | a = x_[alpha] | u_[alpha]| | | | | | |([alpha], [beta])| | | | b = x_[beta] |u_([beta])| |([alpha], [beta], [gamma])| | | | |([beta], [gamma])| | | | c = x_[gamma] | u_[gamma]| . | . | | | . | . | . | . | | | . | . | . | . | | | . | . | . | . | | +---------------+----------+-----------------+--------------------------+------+ in this table, however, ([alpha], [beta]) does not mean u_([beta]) - u_([alpha]), but u_([beta]) - u_([alpha]) / ([beta] - [alpha]); ([alpha], [beta], [gamma]) means {([beta], [gamma]) - ([alpha], [beta])} / (1/2)([gamma] - [alpha]); and, generally any quantity ([eta], ... [phi]) in the column headed "rth diff." is obtained by dividing the difference of the adjoining quantities in the preceding column by ([phi] - [eta])/r. if the table is formed in this way, we may apply the principle of s 12 so as to obtain formulae such as x - a x - a x - b u = u_[alpha] + ----- . ([alpha], [beta]) + ----- . ----- . ([alpha], [beta], [gamma]) + ... (21), h h 2h x - c x - c x - b u = u_[gamma] + ----- . ([beta], [gamma]) + ----- . ----- . ([alpha], [beta], [gamma]) + ... (22). h h 2h the following example illustrates the method, h being taken to be 1 deg.:-- _example 8._ +--------+------------+-------------+-------------+-------------+ | x. | u = sin x. | 1st diff. | 2nd diff. | 3rd diff. | | | | (adjusted). | (adjusted). | (adjusted). | +--------+------------+-------------+-------------+-------------+ | | | + | - | - | | 20 deg.| .3420201 | | | | | | | 162932 50 | | | | 22 deg.| .3746066 | | 1125 00 | | | | | 161245 00 | | 48 75 | | 23 deg.| .3907311 | | 1222 50 | | | | | 158800 00 | | 48 30 | | 26 deg.| .4383711 | | 1303 00 | | | | | 156194 00 | | 47 49 | | 27 deg.| .4539905 | | 1445 47 | | | | | 151857 60 | | 46 00 | | 32 deg.| .5299193 | | 1583 48 | | | | | 145523 67 | | | | 35 deg.| .5735764 | | | | +--------+------------+-------------+-------------+-------------+ to find u for x = 31 deg., we use the values for 26 deg., 27 deg., 32 deg. and 35 deg., and obtain 5 5 4 u = .4383711 00 + ---(156194 00) + --- . ---(-1445 47) + 1 1 2 5 4 -1 --- . --- . ---(-46 00) = .5150380, 1 2 3 which is only wrong in the last figure. if the values of u occurring in (21) or (22) are u_(alpha), u_(beta), u_(gamma), ... u_(lambda), corresponding to values a, b, c, ... l of x, the formula may be more symmetrically written (x - b) (x - c) ... (x - l) (x - a) (x - c) ... (x - l) u = ---------------------------u_[alpha] + ---------------------------u_[beta] + ... (a - b) (a - c) ... (a - l) (b - a) (b - c) ... (b - l) (x - a) (x - b) (x - c) ... ... + ---------------------------u_[lambda] (23). (l - a) (l - b) (l - c) ... this is known as _lagrange's formula_, but it is said to be due to euler. it is not convenient for practical use, since it does not show how many terms have to be taken in any particular case. 14. _interpolation from tables of double entry._--when u is a function of x and y, and is tabulated in terms of x and of y jointly, its calculation for a pair of values not given in the table may be effected either directly or by first forming a table of values of u in terms of y for the particular value of x and then determining u from this table for the particular value of y. for direct interpolation, consider that [delta] represents differencing by changing x into x + 1, and [delta]' differencing by changing y into y + 1. then the formula is u_(x, y) = (l + delta)^x (1 + [delta]')^y u_(0, 0); and the right-hand side can be developed in whatever form is most convenient for the particular case. references.--for general formulae, with particular applications, see the _text-book of the institute of actuaries_, part ii. (1st ed. 1887, 2nd ed. 1902), p. 434; h. l. rice, _theory and practice of interpolation_ (1899). some historical references are given by c. w. merrifield, "on quadratures and interpolation," _brit. assoc. report_ (1880), p. 321; see also _encycl. der math. wiss._ vol. i. pt. 2, pp. 800-819. for j. d. everett's formula, see _quar. jour. pure and applied maths._, no. 128 (1901), and _jour. inst. actuaries_, vol. xxxv. (1901), p. 452. as to relative accuracy of different formulae, see _proc. lon. math. soc._ (2) vol. iv. p. 320. examples of interpolation by means of auxiliary curves will be found in _jour. royal stat. soc._ vol. lxiii. pp. 433, 637. see also differences, calculus of. (w. f. sh.) interpretation (from lat. _interpretari_, to expound, explain, _interpres_, an agent, go-between, interpreter; _inter_, between, and the root _pret-_, possibly connected with that seen either in greek [greek: phrazein], to speak, or [greek: prattein], to do), in general, the action of explaining, or rendering the sense of an obscure form of words or an unknown tongue into a language comprehended by the person addressed. in legal use the word "interpretation" is employed in the sense of ascertaining the meaning of the language of a document, as well as its relation to facts. it is also applied to acts of parliament, as pointing out the sense in which particular words used therein are to be understood. the interpretation of documents and statutes is subject to definite legal rules, the more important of which will be found in the articles contract, statute, will, &c. interregnum (lat. _inter_, between, and _regnum_, reign), strictly a period during which the normal constituted authority is in abeyance, and government is carried on by a temporary authority specially appointed. though originally and specifically confined to the sphere of sovereign authority, the term is commonly used by analogy in other connexions for any suspension of authority, during which affairs are carried on by specially appointed persons. the term originated in rome during the regal period when an _interrex_ was appointed (traditionally by the senate) to carry on the government between the death of one king and the election of his successor (see rome: _history_, _ad init._). it was subsequently used in republican times of an officer appointed to hold the _comitia_ for the election of the consuls when for some reason the retiring consuls had not done so. in the regal period when the senate, instead of appointing a king, decided to appoint _interreges_, it divided itself into ten decuries from each of which one senator was selected. each of these ten acted as king for five days, and if, at the end of fifty days, no king had been elected, the rotation was renewed. it was their duty to nominate a king, whose appointment was then ratified or refused by the _curiae_. under the republic similarly _interreges_ acted for five days each. when the first consuls were elected (according to dionysius iv. 84 and livy i. 60), spurius lucretius held the comitia as interrex, and from that time down to the second punic war such officers were from time to time appointed. thenceforward there is no record of the office till 82 b.c., when the senate appointed an _interrex_ to hold the _comitia_ which made sulla dictator (appian, _bell. civ._ i. 98). in 55, 53 and 52 _interreges_ are again found, the last-mentioned being on the occasion when pompey was elected sole consul. the most noteworthy use of the term "interregnum" in post-classical times is that of the great interregnum in german history between the death of conrad iv. (1254) and the election of rudolf of habsburg (1273). see germany: _history_. interstate commerce. the phrase "interstate commerce," as used in the united states, denotes commerce between the citizens of different states of the union. the words "interstate" and "intrastate" are not found in the constitution nor, until comparatively recently, in decisions of the courts or in legislative acts (probably being first used officially in 1887 in the interstate commerce act). the constitution of 1789 uses the phrase "commerce among the states," and the first official decision interpreting the phrase says that "it may very properly be restricted to that commerce which concerns more states than one" (chief justice marshall in _gibbons_ v. _ogden_, 9 _wheaton_ 194). commerce among the states is there distinguished from "commerce which is completely internal, which is carried on between man and man in a state, or between parts of the same state, and which does not extend to or affect other states." it was declared (_lehigh_ case, 145 _u.s._ 192) that commerce between two persons in the same state is not interstate even when there is a temporary deviation to the soil of another state; but later (_hanley_ case, 187 _u.s._ 617, distinguishing the _lehigh_ case) it was declared that as to transportation, such commerce is interstate. the courts have interpreted commerce to denote not merely a mutual selling or traffic, but as "a term of the largest import," including intercourse for the purposes of trade in any and all its forms (_gibbons_ v. _ogden_, 9 _wheaton_ 194, and _welton_ v. _missouri_, 91 _u.s._ 280). thus have been included not only the actions of trading, navigation, transportation, and communication, but also the instruments and agents employed, including even telegraph messages and, in the extremest cases, lottery tickets.[1] the decision of the question where federal control of interstate traffic ends and state control begins has been one of great practical difficulty. in general it has been held that whenever a commodity begins to move as an article of trade from one state to another, commerce in that commodity between the states has begun. mere intention to ship goods does not make them subjects of interstate commerce, but they must actually be put in motion or committed to the carrier for that purpose (_coe_ v. _errol_, 116 _u.s._ 517). as a practical guide in deciding when state control should be resumed, the court as early as 1827 (_brown_ v. _maryland_) laid down the "original package rule," that the taxing power of the state should begin when the original package in which the goods had been imported into the state had been broken up or sold. the injustice of allowing goods to be held thus, for long periods escaping local taxation, led to a modification of the rule in 1868 (_woodruff_ v. _parkham_, 8 _wall._ 123), and such goods after reaching their destination may be taxed as property in common with other property in the state.[2] _reason for federal control of interstate commerce._--immediately after the close of the war of american independence in 1783 appeared the separatist tendencies and local jealousies usual in a confederation. the congress of the confederation had no power to levy tariff duties or to regulate commerce between the states, and the separate states freely and recklessly exercised their rights in this matter. though commerce at that time was comparatively unimportant, the results of this restrictive policy were most unfortunate. the annapolis convention of 1786 was called by the virginia legislature to take into consideration the trade of the united states and to consider how far a uniform system in their commercial relations might be necessary to the common interests and their permanent harmony. this conference resulted in the call of the philadelphia convention of 1787, which framed the present constitution. chief justice marshall, in one of the early cases on this subject (_brown_ v. _maryland_, 12 _wheaton_ 419, in 1827), said in words often since quoted: "it may be doubted whether any of the evils proceeding from the feebleness of the federal government contributed more to that great revolution which introduced the present system than the deep and general conviction that commerce ought to be regulated by congress." every year has increased the importance of the congressional power of regulating commerce. at the time of the adoption of the constitution, each neighbourhood supplied nearly all its needs by its own industry, but improving means of transportation and communication have multiplied the commercial ties between the citizens of the various states. this change went on slowly until 1830, more rapidly between 1830 and 1860, and at an ever-hastening pace after the civil war. until 1824 no case involving directly the consideration of this power reached the united states supreme court. from 1824 to 1840 the supreme court decided an average of one-third of a case a year; from 1841 to 1860, an average of three-fourths of a case; from 1861 to 1870, an average of one case; from 1871 to 1880, an average of nearly six cases; from 1881 to 1890, an average of more than seven cases; and from 1891 to 1900, an average of more than ten cases. the decisions have not been entirely uniform, and there were some decisions too contradictory to be explained by any ingenuity. the supreme court itself has said (_fargo_ v. _michigan_, 121 _u.s._ 230) that "it may be admitted that the court has not always employed the same language, and that all of the judges of the court who have written opinions for it may not have meant precisely the same thing." though in the period just preceding the civil war the doctrine of states' rights tended to weaken somewhat the federal power, the broad outlines of the interpretation by chief justice marshall laid down in 1824 in _gibbons_ v. _ogden_ remain to-day almost undimmed. _interstate commerce in the federal constitution._--freedom of trade, without discrimination, between the citizens of all the states was in the main ensured by one brief sentence, usually called the "commerce clause" of the federal constitution:--"the congress shall have power ... to regulate commerce with foreign nations, and among the several states, and with the indian tribes" (art. 1, sec. 8, clause 3). hardly less important is the power "to make all laws which shall be necessary and proper for carrying into execution the foregoing powers, and all other powers vested by this constitution in the government of the united states, or in any department or officer thereof" (art. 1, sec. 8, clause 18). to the same end of freedom of commerce, congress is limited in that "no tax or duty shall be laid on articles exported from any state," and "no preference shall be given by any regulation of commerce or revenue to the ports of one state over those of another; nor shall vessels bound to or from one state be obliged to enter, clear, or pay duties in another" (art. 1, sec. 9, clauses 5 and 6). directly and by implication, congress was granted a number of other powers over commerce, in that it may coin money, establish uniform laws of bankruptcy, establish post-offices and post roads, regulate weights and measures, exercise admiralty jurisdiction (now interpreted to extend to all public waterways accessible to the traffic of more than one state), grant patents and copyrights, and use the power of taxation to protect, repress or even destroy the agencies of commerce (e.g. state bank notes). but these powers can be exercised only in ways which favour and make free the intercourse among all parts of the nation. even if the commerce clause had been omitted from the constitution, a large part of its object would have been attained by certain prohibitions upon the states as follows: "the citizens of each state shall be entitled to all privileges and immunities of citizens in the several states" (art. 4, sec. 2). "no state shall, without the consent of the congress, lay any impost or duties on imports or exports, except what may be absolutely necessary for executing its inspection laws; and the net produce of all duties and impost, laid by any state on imports or exports, shall be for the use of the treasury of the united states, and all such laws shall be subject to the revision and control of the congress" (art. 1, sec. 10, clause 2). "no state shall, without the consent of congress, lay any duty of tonnage" (art. 1, sec. 10, clause 3). thus by threefold measures of precaution was ensured domestic freedom of trade from every point in the land to its farthest frontiers. _negative working of the commerce provisions._--for nearly a hundred years these provisions were important only in their negative effects of preventing the states from granting special privileges to their citizens or taxing unequally the citizens of other states. the decision in 1824 of _gibbons_ v. _ogden_ stopped the attempt of the state of new york to grant the monopoly of steamboat traffic on the waters of that state. had the clear and unequivocal opinion in that case been different, local ingenuity doubtless would have devised a multitude of discriminations. "the power to tax involves the power to destroy," and ever since the decision of _mcculloch_ v. _maryland_ in 1819 it has been held that no agencies created by the federal government, such as banks or legal tender notes, are subject to state taxation, and the rule has also been laid down repeatedly by the supreme court (for the first time in 1886) that no burden can be laid upon the act of taking goods into or out of the state, of soliciting sales, or of delivering goods even though the tax is without discrimination as between the state's own citizens and others; that is, interstate commerce "cannot be taxed at all" (_robbins_ v. _shelby county taxing district_, 120 _u.s._ 489).[3] federal control of interstate commerce has been interpreted by the courts to be exclusive of any control by the states. this is not self-evident in the clause, "congress shall have power to regulate commerce among the several states." over some other subjects the power of the federal and state governments is concurrent, the state being able to act until congress enacts some conflicting legislation. although the early decisions suggested that the power of congress was exclusive, yet for nearly a century no positive decision was rendered and no positive action was taken by congress. between 1870 and 1886 the states made great progress in the regulation of railways on the assumption that until congress had acted the states were free to act. the question was put beyond doubt in a series of decisions establishing the principle that the non-action of congress indicates its will that commerce shall be free and untrammelled and that the states cannot interfere either through their police power or their taxing power.[4] _positive federal regulation._--though the regulation of interstate commerce up to the civil war was mainly negative, some positive actions of the federal government had indirect effects on commerce, as, for example, the coinage of money, the establishment of post-offices, the charter of the first and second united states banks, and the charter of the pacific railroad. the power to do these things was conferred by the constitution in some cases directly, in other cases by implication in that any means appropriate to lawful ends might be employed (as in case of charter of the united states bank, _mcculloch_ v. _maryland_). from 1850 to 1862 the federal government had made numerous land grants in aid of railways, but always to the states, not directly to the corporations, and it had never until 1862 granted a charter to a railway, canal, turnpike or transportation company. in 1866 congress passed an act authorizing railway companies whose roads were operated by steam to carry passengers, freight, &c., "on their way from any state to another state and to receive compensation therefor and to connect with roads of other states so as to form continuous lines for the transportation of the same to the place of destination."[5] this act, so vague and general in its terms, had very little effect, though it has been the occasion of considerable litigation to determine its influence upon existing police laws of the states. in 1884 congress established the bureau of animal industry for preventing the exportation of diseased cattle and for the extirpation of disease among domestic animals. this had little significance at the time for interstate commerce, its purpose being to meet the objections of foreign countries to the importation of american meat. in 1887 was passed the interstate commerce act, providing a national commission to supervise interstate railways. in 1888 was passed an arbitration act, replaced in 1898 by an act which provides that in case of disputes between common carriers subject to the interstate commerce act and their employees, conciliation shall be tried, and, in case this should fail, indicates the methods that may be used for the voluntary submission of the dispute to a board of arbitration. in 1890 was passed the sherman anti-trust act, making illegal every contract and combination in restraint of trade or commerce among the several states or with foreign nations. in 1893 a safety appliance act, the administration of which was put into the hands of the interstate commerce commission, promoted the safety of employees and travellers, and required the roads engaged in interstate commerce to equip their cars and locomotives with automatic couplers and brakes. in 1895 was prohibited the interstate carriage of condemned carcasses of animals, and of lottery tickets (see above reference to the interpretation of the lottery act), in 1897 of obscene literature, and in 1900 of game killed in violation of state laws. in 1901 carriers engaged in interstate commerce were required to make full reports of all accidents to the interstate commerce commission. in 1902 was prohibited the interstate carriage of dairy products falsely labelled or branded as to the state or territory in which produced, and in 1903 the secretary of agriculture was empowered to establish rules concerning importation and transportation of live stock. in 1903 the bureau of corporations was established with power to investigate the conduct of corporations engaged in interstate and foreign commerce, excepting common carriers subject to the interstate commerce act. in 1903 the interstate commerce act was amended by the elkins act, making much more difficult the granting of rebates. in 1905 the president was authorized to grant medals of honour to persons who by their daring save life or prevent accident on railways. in 1906 the interstate commerce act was amended in important particulars (specified below). in 1906 were passed pure food laws, greatly enlarging the duties of the department of agriculture in reference to inspection of foods prepared for interstate commerce. _the interstate commerce act._--the period of positive action by congress in the regulating of interstate commerce practically begins, therefore, with the enactment of the interstate commerce act of february 1887, the outcome of fully seventeen years of agitation and discussion. the law was modelled in large part upon english acts. it applied to common carriers wholly by railway, and partly by railway and partly by water when both are used under a common arrangement for continuous shipment; forbade unjust discrimination and undue and unreasonable preference; made it unlawful to charge more for a shorter than for a longer distance over the same line in the same direction, the shorter being included within the longer distance (though a carrier might be freed by the commission from the working of this provision); and forbade pooling and division of earnings. the administration of the law was entrusted to a commission of five members, appointed by the president. from this act much was expected, but eighteen years of its operation gave as net results little more than a greater uniformity of railway accounting and much better understanding by the public of the nature of the railway problem. discrimination and secret rebates continued. the anti-pooling clause (pretty generally recognized by the well-informed to be a mistake) prevented open but not secret agreements between carriers, and probably hastened the movement toward consolidation. the long and short haul clause was made meaningless by the judicial interpretation that any competition, even that of other carriers subject to the act, justified the railway in charging more for a shorter than for a longer haul. the effectiveness of the commission was destroyed by the judicial decision that it had no power to fix rates for the future. until 1897, the commission, when it adjudged a rate unreasonable, usually declared what rate was reasonable, and directed the carrier to reduce the rate by a given date to the designated maximum. of 135 orders made in decisions rendered in the first ten years of the commission, 68 prescribed a maximum rate for the future. in 1897 it was finally decided in the _cincinnati freight bureau case_ (167 _u.s._ 479) that congress had not conferred upon the commission the power to prescribe any rate for the future. the court said that congress might fix the rate itself or authorize a sub-tribunal to do so, but that congress had not yet given that authority. the need of further legislation had been felt from the beginning by many, and after 1903 the agitation became very active. the position taken by president roosevelt in his message to congress in 1904 made the amendment of the interstate commerce act the principal political issue before congress in the sessions of 1905 and of 1906. after the most remarkable senatorial debates heard at washington in years, followed with close interest by the country, a number of amendments became law on the 29th of june 1906. the act was strengthened to a degree hardly expected by the most earnest advocates of revision. a number of minor changes made in the light of experience were: increasing the number of commissioners to seven and their pay to $10,000; facilitating procedure and the taking of evidence; requiring thirty days notice of a change of rates; requiring appeal from the commission's decision to be taken within thirty days; empowering the commission to establish joint rates and to order switches to be built. the following are generally thought to be still more important changes: (1) including within the application of the act pipe lines (particularly for oil), express and sleeping car companies, and all the facilities and services in connexion with goods transported; (2) giving publicity to railway business by empowering the commission to prescribe all forms of accounts and to examine the books at all times, and by forbidding any other accounts or memoranda to be kept by the companies; and (3) empowering the commission to prescribe reasonable maximum rates to take effect within not less than thirty days and to continue not over two years unless set aside by the courts. _the anti-trust act of 1890._--the growth of large corporations with some degree of monopoly power, the so-called trusts, had called forth in a number of the states anti-trust laws before 1890. when it became evident that the states were not succeeding in dealing with the problem, public sentiment found expression in the sherman anti-trust act, approved on the 2nd of july 1890. this act declared illegal and criminal, punishable by fine or imprisonment or both, every contract in restraint of trade or commerce among the several states or with foreign nations. the statute thus changed the common law wherein such contracts were merely unenforceable but not criminal. this act was at first construed by the supreme court as applying to any contract in restraint of interstate commerce, whether reasonable or unreasonable (_trans-missouri freight association_, 166 _u.s._ 331), but later, in 1905 (_stock yards case_, 25 _supreme court reporter_ 276) it was held that the act did not apply to agreements for the better conduct of business which incidentally affected interstate commerce.[6] the act has been interpreted to apply to transportation (_freight association case_, 166 _u.s._ 290, and _northern securities case_), with results felt even by some of the advocates of railway regulation to be unfortunate. it applies to unlawful combinations of manufacturers to divide the territory and regulate the prices (_addyston pipe trust case_, 175 _u.s._ 211). in the sugar trust case (1895 _u.s._ v. _knight co._ 156 _u.s._) it was declared that the statute did not apply to a manufacturing company which had acquired nearly complete control of the manufacture of refined sugar by means of the purchase of stock of other refining companies. the attorney-general submitted to the senate, in june 1906, a statement of the results of all suits instituted by the department of justice under the anti-trust law, the interstate commerce act and the elkins act, in the period from 1887 to june 1906 inclusive. thirty-six suits were still pending; of the 250 which had been disposed of in some manner 186 ended in dismissal, non-prosecution or acquittal, and 64 were successful in securing in whole or in large part the object of the suit (in 30 cases conviction, in 34 cases the granting of a petition or an injunction, &c.). in addition to these results of federal efforts to regulate industry must be counted the cases in which carriers complied with the orders of the interstate commerce commission without suit; but even then the total by 1906 was somewhat meagre. the establishment of the bureau of corporations in 1903, and the considerable extension of the powers of inspection of the department of agriculture are recent changes of which the results cannot yet be fairly judged. the aim of the bureau of corporations is to ensure publicity in the management of corporations engaged in interstate and foreign commerce. the first commissioner, mr james r. garfield, showed much activity in pursuing the purposes of the act, and published informing reports upon the beef trust (1905), and upon the standard oil company (1906). but the effect and possible extension of federal interference became from this time burning political questions of far-reaching importance of too recent a date to be dealt with historically in this article. see also the _annual reports_ of the interstate commerce commission since 1887, and decisions; prentice and egan, _the commerce clause of the federal constitution_ (chicago, 1898); _reports_ of the commissioner of corporations on the beef industry (1905), on the transportation of petroleum (1906); w. z. ripley (ed.). _trusts, pools and corporations_ (1905), containing leading cases and analyses of the voluminous "trust" literature; f. n. judson, _the law of interstate commerce and its federal regulation_ (chicago, 1905); beale and wyman, _railroad rate regulation_ (boston, 1906); frank hendrick, _the power to regulate corporations and commerce_ (new york, 1906), favouring less of new legislation. (f. a. f.) footnotes: [1] the lottery tickets were included only by a divided court (_lottery cases_, 188 u.s. 321) four judges emphatically dissenting. the moral issue doubtless influenced a decision so difficult to reconcile with other opinions of the court, which otherwise had held regularly that commerce involves the physical movement of persons or things and does not include the contractual relations between citizens incident to commercial intercourse. not all things incidental to commerce are included in it, and it has been held that the following are not included: bills of exchange (in 1850, _nathan_ v. _louisiana_, 8 _how._ 73), trade marks (in 1879, _trade mark cases_, 100 _u.s._ 82), insurance (in 1869, _paul_ v. _virginia_, 8 _wall._ 168), and manufacturing (in 1895, _u.s._ v. _knight co._, 156 _u.s._ 1). in the last-named case, which concerned a combination of sugar refineries controlling a large proportion of the product of the country, it was said that commerce succeeds manufacture and is not a part of it. the relation of the manufacturer to interstate and foreign commerce being thus only incidental and indirect, the business is subject to state control. by a series of decisions the transportation of persons has been decided to be commerce. (in 1848, _passenger cases_, 7 _how._ 283. in 1867, _crandall_ v. _nevada_ 6, _wall._ 35. in 1875, _henderson_ v. the _mayor of new york_, _92 u.s._ 259, &c.). [2] the question arose with reference to the police power of the state in those states prohibiting the liquor traffic, and in 1889 it was held (leisy _v._ hardin) that, in the absence of legislation by congress, the right to sell goods taken into a state was unrestricted. this made it impossible for a state to exclude the importation of liquors to be sold within its territory, but this difficulty was remedied by the wilson original package bill of 1890, which made liquor subject to the police powers of the state to which it was carried. [3] however, a very important distinction is drawn between taxing the commerce and taxing property employed in commerce. with the increase of interstate commerce, the states have been hard pushed to find sources of revenue adequate to their increasing needs. the courts, therefore, have sought to draw a line between taxes on the privilege of carrying on interstate commerce and taxes on the property employed in carrying on such commerce as a part of the general body of property in the state. thus it has been held in the case of _state freight tax_ (1872, 15 _wall._ 232) that a state could not lay a tax on freight transported from one state to another, and yet the same year the court held in _state tax on gross receipts_ (15 _wall._ 284) that a tax was valid when laid upon the receipts of railways organized under the laws of the state, as upon a fund which had become incorporated with the general mass of property. this latter decision was by a divided court (three of the nine judges dissenting), but it has since been frequently confirmed. the tax on gross receipts of all railway companies doing business in the state has been supported when levied in proportion to the mileage within as compared with the total within and without the state (_erie ry._ v. _pa._, 21 _wall._ 492). this so-called "unit rule," as applied either to gross receipts or to the entire value of an interstate railway, has been upheld in a number of decisions. the method of taxation by gross receipts, however, has not tended to increase of late, but the unit rule, as applied to _ad valorem_ taxes on property, is more and more being applied. every case involving the distinction between a tax on commerce and a tax on property employed in commerce presents its own difficulties, yet a practical way is thus found to prevent discriminating action by the several states, while leaving to them adequate sources of revenue. [4] 1873, _state freight tax_, 15 _wall._ 232; 1887, _robbins_ v. _shelby county taxing district_, 120 _u.s._ 489; _wabash r. r. company_ v. _illinois_, 118 _u.s._ 557. the last-named case arose out of the attempts of the state of illinois to prevent discrimination between two shippers, both being its own citizens and within its own borders, one of whom was being charged more than the other for a shorter shipment on the same line and in the same direction, from a point outside the state. the court, applying the established definition of interstate commerce with verbal formality of logic, decided that the state could do nothing, for even in such a case all regulation of interstate commerce, from the beginning to the end of a shipment, was confided to congress exclusively. thus a clause whose clear purpose was to prevent one state from burdening unequally the citizens of other states was successfully invoked by a private corporation to forbid the state securing equality of treatment for its own citizens as regards such parts of shipments as lay within its own borders. most railway traffic was by this decision declared to be subject to legislation by congress but congress had not acted. the impossibility of this situation was so evident that the interstate commerce act, long under discussion, became a law a few months later. [5] this was probably aimed at the discriminating between new york and philadelphia (see speech of charles sumner on the railroad usurpation of new jersey in u.s. senate, february 14, 1865). [6] in the northern securities case, justice brewer, who had concurred in the opinion in the trans-missouri freight association case, took occasion to say that while he still believed the former case had been correctly decided, he thought that the reasons given for the judgment were in some respects faulty, and that the ruling should have been that the contracts there considered were unreasonable restraints and as such were forbidden by the act.