﻿200 ON SOME APPLICATIONS OF THE METHOD 



Re CT ardin ff Fx, Fy, and Fz as the first differential coefficients of Vx, Vy, and Vz, 



we obtain from the series Fx Fx n , Fy Fy n , Fz Fz u , by using (5), 



Vx, Vy, and Vz for t*=*(n — i) x. As we know x, y, and z for the same epoch, we 

 can compute the elements of the orbit at t = (n — i)x, and the differences between 

 them and those at t = are their perturbations from 2 = to t=(n — £) x. When 

 n x is very large, which will depend on the heliocentric motion of the disturbed body, 

 it will not be safe to use logarithms of five figures to find Fx, Fy, and F z, and the 

 labor of computing them is thereby much increased. 



In order to avoid this difficulty, if we denote by S n the whole perturbation of the 

 quantity to which it is affixed up to the moment t = nx, so that 8x n is the difference 

 between the actual value of x„ and that which it would have had if from the time 

 Z = the sun's force alone had been exerted upon it, we may suppose the whole 

 period to be divided into portions such that from t = to t' = 0, from t' = to f =0, 

 &c, the squares of Sx, Sy, and Sz, and their products with m, m", &c, are insensible; 

 we then have from (36) 



WIN — #.{?-§£) -,4 P* f -F, (£-#£)-,* Fl.«F«(£-t£)-,e5 

 by which Sx, 3y, and Sz may be determined from t = to t' = 0, independently of 

 the exact values of x, y, and z. For although the unknown quantities 5a:, Sy, and S z 

 enter the expressions (39), the coefficient £ may be made as small as we please, it being 

 of the order of the square of the heliocentric motion of the disturbed body in one in- 

 terval. So that, by taking z sufficiently small, we make the first terms of FSx, 

 FSy, and FSz very small compared with ft A, fiB, and (i C, and this will be the case 

 for several of the first intervals. 



At t = 0, 5x„ = 0, V3x = 0, FSx = — fiA , 



(40) «5j/„ = 0, VSy = 0, FSy a = — nB , 



3z„=0, V9e — ^> F5z = — f,C , 



(10) gives J' S x = — i ft A a , neglecting d\, and thence 



g Xi = — ipA t , 8 y x = — i fi B , dz l = — i t iC . 



With these we obtain FSx,, FSy,, and FSz noticing that i 8r* = xSx + ySy + zSz. 

 After correcting Sx„ Sy,, and 8 z, as far as necessary for the differences of FSx, FSy, 

 and FSz, the computation of the rest of the series will be direct, and conducted on the 

 same principles as have already been detailed in the preceding sections. From Sx, Sy, 

 and Sz, we pass to the perturbations in geocentric right ascension and declination by 

 means of (35). 



FSx, FSy, and FSz being the first differential coefficients of 8 Vx, 8 Vy, and 8 V z, 



