264 Variations of the Arbitrary Constants in Elliptic Motion. 



the result with (D), having regard to (95), we shall have ^t? = 



2rd6r-{-dr.6r ^ ^ * ^ dR 



^—j -i-3affndtdR-\-2afndtr-^} (I), if M is not sup- 



posed =1, we must divide the two last terms of the numerator of 



m' 

 (I) by M, since tTx enters as a factor into R, or if there are several 



m' m" , 



disturbing bodies as m', m", he, then the factors ^j ttfj &;c., will 



enter into the several parts of R which depend on the disturbing 

 bodies m', m", &;c. severally ; hence (I) will become the same as 

 the formula (Y), given by La Place at p. 258, Vol. I of the Mec. 

 CeL, which was found by him in a very different manner from the 

 above. 



Again, substituting r- for r'^ in (64), then taking the hyperbolic 

 logarithms, we get log. dv = log. dt + log. c — 2 log. r, whose varia- 



'()c 2Sr' 



tioh (since dt = const.) gives d^v=^ I — — j dv, and by taking 



(Sc 25r\ 

 the integrally =/(— — — jdv, (K) ; which is the formula given 



by Pontecoulant at p. 474, Vol. I of his Systeme du Monde, and 



by Mrs. Somerville at p. 296 of her Mechanism of the Heavens, 



but their methods of investigation [which are exactly the same,^ are 



by no means so simple as the above. 



dv 



rdv c l^^-^di'^^^] 

 By (64) --77 = - which reduces (K) to 6v=f\ jdv, 



. dv 



(96), since we have c=v a(l— e^), n'^a'^^X^ dv— Z^^ndt, (96) 



na dv I dv \ 



is easily changed to <5t;= ,- ^ ^f^t V'^^^-^ ^r5r jndf^, (L) ; 



which appears to us to be a better form for calculating ()V than that 

 given by (K). 



By taking the plane of the primitive orbit of m for that of the 



dH Mz dR 

 plane x, y, the last of (5) which is -^ + 7^ + ^ =^' (^)' ^"^ 



enable us to find z, and then by (63), since r'=r, we shall get s= 

 -} or if we please we may put rs for z in the two first terms of (M), 



