are performed they vanish of themselves. The complementary function in this case is 6--("Hy+c8) o(y, z) = (y – bx, z - cx); therefore, effecting the operations indicated, If we integrate with respect to t we find == Bat (x) = (2) + at dP(x), a’t d'P(x) + &c. dait T1.2 dxt If we integrate with respect to a we shall have two arbitrary functions of t, since the differential with respect to the former variable is of the second order. Writing the equation in the form d’: 1 dz da" a dt we may divide it into two factors Id 11 d 11 d) Id aldt daa Whence <=C4H). p(t) +e-Caf. Vo(t); or, if we put p(t) + y () = F(t), and (c) * {$(t) – 4 (0)} = f(t), this may be put under the form * = F(t) + mai midF(t) 1 201 l F(t) Z I a 1.2 dt a" 1.2.3.4 dt" 1 zi df(t) 1 d?f(t), +xf(t) + 4 ? + &c. ' a 1.2.3 d t ' a 1.2.3.4.5 dt = + = It seems anomalous that the same equation should admit of two solutions differing so essentially in character that the one contains two arbitrary functions and the other only one: but the following considerations may serve to explain the difficulty. Since by Maclaurin's theorem any function of a variable may be expressed by means of its differential coefficients, taken with respect to that variable, we know the function if we can determine its successive differential coefficients. Now from the equation dz ďx 1 dt = a da we can determine the values of all the differential coefficients with respect to t, when t = 0, if we know the value of x when t = 0. This therefore is the only undetermined quantity in this case, and it corresponds to the arbitrary function ® (). But from the equation ď % idx dvadt' we can only, from the value of x when x = 0, determine the values of the alternate differential coefficients : and in order to determine the others we must also know the value of "da when a = 0. Therefore in this case there are two indeterminate quantities corresponding to the arbitrary functions F (t) and f (t). This is the equation for determining the linear transmission of heat in an infinite solid. Fourier, Traité de la Chaleur, p. 471 and p. 509. Whence <=ff(x). This is the equation for determining the motion of heat in a ring. Fourier, Ib. p. 266. The operating factor in this case may be decomposed into two, and the equation then becomes id did di (1 Whence x = f***? $(2) +e* 14(1x); or x = Q(x + at) + y (x – at). This is one of the most important equations in the application of Mathematics to Natural Philosophy, being that which results from the investigation of the motion of vibrating chords, and of the pulses produced by a disturbance in a small cylindrical column of air. (7) Let dady ,2 = xy. The complementary function in this example is the same as the integral of the last, and the result of the inverse operation on xy will best be found by expanding in ascending powers of C ), when it is easy to see that all the terms after the first may be neglected. We find accordingly The solution of this is - + cos Idyl Compare this with Chap. iv. Sect. 1, Ex. (8). (9) The equation ta? -24 de dy " d at = 0, may be put under the form d 2 dyl so that the two factors are equal. Hence, integrating with respect to , *=**** da".0 = 2**{op() + 4 (9)};. or x = xo (y + ax) + y (y + aw). - - - - gas)} a + ) = = 0. omes The effect of the second factor on the second side of the equation will be simply to alter the function of y, and as that is arbitrary we may leave it as it stands, so that we have, on adding the complementary function due to the second factor, = -+ (4) + “, (4), or x==2018 (y + a x) + y(y – ax). Euler, Calc. Integ. Vol. 111. p. 210. d28 de de (11) Let +6 + abx = V, |