whence, dividing through successively by dx and dy, and we have the ratio of the partial variations of u and of x, and of u and of y, and bracketing them to indicate that they are partial, we have vary when (du) is calculated, and a does is calculated, we must bear in mind that dr de du dy are to be calculated on the supposition that dy = 0; dx' dx dr de and that dy' dy are to be calculated on the supposition that dx=0. Let us introduce these conditions; from (178) we have successively do and dr between these two equations on this supposition, we have and if we substitute these quantities in the expressions above du "), the resulting expressions will be the equi dy and dy when x and y are replaced by their equivalents in terms of r and 0. It is unnecessary to express the values in their full length, because it is more convenient to work each example with its own particular formulæ. dR 107.] Ex. 1. To transform (1) and dR dy into their equi valents in terms of r and 6, when ar cos 0, y = r sin 0. In this problem there is implied a function, R = f(x, y), which becomes, when x and y are replaced by their equivalents, To calculate da, dy must be equal to 0; whence, eliminating de and dr in turn from (187), we have dx dr cos 0-r sin 0 do, and therefore 0 = dr sin 0 + r cos 0 d0 ; Similarly to calculate dy, dx = 0; wherefore, by means of Hence we have two transformations useful in the Planetary Theory, viz. dR dR (dR), = dy Ex. 2. It is required to transform into its equivalent in terms 108.] The general formulæ for the transformation of second and higher partial differentials and derived-functions, which are analogous to those of Art. 107, are evidently very long; and as their discovery is easy, it will be sufficient for our purpose to give one or two examples wherein particular and the most common forms of them appear. Ex. 1. To transform (d2) + d2v dy2 into its equivalent in terms of r and 6, where x = r cos 0, y = r sin 0. By the process of the preceding Article and (188), we have differentiating (191), and bearing in mind that are functions of r and 0, we have (dv) and (dv) The results (193) and (194) are no other than particular cases of equation (100), Art. 80, when the right substitutions are made, and consistently with the independence and equicrescence of the variables. Also by a process similar to that by which (d) and have been found, it may be proved that dx2 d2v dy2 sin cos 0/d2v до 2 cos 20 (dv (dx dy) = sin 0 cos 0 ( dr de) (day) (196) r.2 lent in terms of r, 0, and 4, when arsin cos p, y=rsin 0 sino, 2 = r cos 0. It is convenient in this example to introduce a subsidiary quantity p, so that d2 1 1 1 dv = ((xx) + 1 = (day) + /= (day) + (dr) + (dr); (200) dr2 Ρ therefore substituting for p from (197) we have, To reduce this farther let cos 0μ; therefore cos-1μ; and = r2 (d2x) + 2r (dv) = r (dry), dr2 so that (201) becomes dr I may observe, see Ex. 3, Art. 102, that the given differential equation expresses a property of V= {(x − a)2+(y—b)2 + (≈ —c)2} − §. If in this we replace x, y, z by their values given in (197), we have PRICE, VOL. I. B b |