﻿A KEW METHOD FOR CORRECTING A PLANET's ORBIT. 471 



If now we vary J^^ we shall have similar equations, in which, after performing the 

 subtraction indicated above, we shall have, in place respectively of — fi, ffi, 0, as 

 known terms, the values^, 0, (j.^; and as unknown quantities, the first derivatives of 

 the elements with respect to z/g. 



18. Having now these derivatives with respect both to Ji and z/a, we can for any 

 other time easily compute, by formula; (11), (19), using the former as modified by (13), 



J. = l5L,-1^31ogp + (^;+f-smE-3^.tan<p)j^ + ^-(^-£--l)a,, (36) 



the values of D^^ z, D^, w ; and as, by (26'), 



the value (which we do not know) of 8 Z will not affect 7f ; and the coefficients D^^ 6 , 

 D^j 6 are, with regard to almost all planets, of such slight amount that they cannot 

 increase the weight of our solution much over j^-^ part, as has been shown in Mr. G. 

 P. Bond's Memoir on " Equivalent Factors," assuming the modulus of D^^ 6 as about 

 ^^, which is as much as it can often be with regard to five sixths of the asteroids. 



19. We have thus, if at least two positions besides those to be exactly satisfied are 

 given, the means of determining what changes the assumed z/j, z/g must undergo to 

 satisfy all the geocentric places we have, with almost if not quite the accuracy of a 

 least-square solution. After this is done, it only remains to compute from the variations 

 of ^1, z/g the changes which the orbit-plane must undergo, in order to pass exactly 

 through the positions to which they refer. This is done by combining (20) and (34). 



"We have 



S Zi :^ s'm $1 5 Ji = r, sin (i-, -\- a>) . S i — r, sin i cos (c, -|- w) . (5 /> ; 

 5 Zj = sin ^2 3 z/j = r, sin (cj -[-<")•'' ' — ''2 ^'^ ' <^o* (^'2 ~l~ ^) 



The readiest practical way to obtain 8 i, 8 SI will be to form these equations and 

 solve them numerically. 



After this is done, we shall find also, by the equations in which D^^ , L , &c. occurred, 

 the values of 5 L , 5 log jj, 8 cp, 8 x', and the resulting changes in the usual elements 



will be, 



(5 L = i5 L, + (1 — cos i) 5 P.; 



3 n = (5 ;f 4- (1 — cos i) 8 P.; 

 d log a = 3 log /> -(- 2 tan 9 . 3 9 sin 1" ; 



(5 ,u = — } fid log p — 3 ju tan q> . d tp sin 1" . 



It is to be noticed that all the variations, those of Ji, z/a, and log p inclusive, will 

 be expressed in seconds ; so that we must add to their logarithms the log 4.685574:9 — 

 10, or must divide them by 206264.8 to reduce them to the unit required. 



■ S SI;) 



(38) 



