﻿198 ON SOME APPLICATIONS OF THE METHOD 



When Vx„ is varied, we have from (24), 



(28) K = 0, VSrl = 2x dVx , Fdr> = 4Vx SVx = - <S (?f ), 



(29) and for any other epoch excepting t = Q, F8r* = — -Sr 2 — S( 2 -£); and since for all 

 values of t, 4 Vx SVx = — S(^) is constant as before, and £ is already known, 5r 2 

 will follow from equations (27) and (28), when Vx is varied, precisely as from equa- 

 tions (26) and (27) when x is varied. 



Having thus found S r 1 from £ = to l = nt, it remains to determine 8x, Sy, and 

 Sz for the same period. 

 From (25) we find 



(30) F8x = Fx{*±-$ 3 *), F8y^Fy(^-^^), Fd z = Fz (^- f !J), 



in which all the quantities are known excepting 5x, fly, and 5r. When x is varied, we 



(3l) haVe ***** V5x =0, F i x. = F* (£-§'£). 



We may, therefore, find approximate values of Sx r and 8 x„ which give accurate values 

 of F8x l and FSx 2 , which may be used to find 8x 3 , and so on, correcting for the differ- 

 ences of F8x as far as is necessary, and we thus obtain Sx from * = to t = nz. 

 Again, when x is varied, 



3y =0, V8y =0, F8y = — Fy ^; 



T 

 ( 32 ) A t 



8*0 = 0, Vaz t = 0, Fdx § = — F%,t'£i 







from which we proceed to find Sy 5z„ &c, which will be small compared with 

 Sx for several intervals. 



When V x is varied, we have, for t = 0, 



8x = 0, V8x = 8Vx , FSx = 0: 



(33) *y. = 0, F*y,, = 0, F8y =0, 

 8z =0, V8z = 0, F8z t = 0; 



and at any other epoch the equations (30), by which Sx, Sy, and Sz may be found 

 from t = to / = »i. 



When 2/ and Vy , z and Fz , are varied, the operations are precisely analogous to 

 those employed for S x and S Vx . 



Finally, the correctness of each series is tested by putting at t = n r 



(34) $8r l =x8x J r -y8y-\-z8z. 



The changes of a and 6, the geocentric right ascension and declination, are, x 

 and i being the geocentric coordinates, 



cos. a . 



sin. 8 a = —j — (3 y cos. a — Sx sin. a) ; 



(35^ - . „ sin. 5 <J ,. , . , 



v ' sin. 8 6 = (0 z cot. 6 — 8 y sin. a — 5 i cos. «). 



z 



The equations of condition are then to be formed and solved as usual. 



