I 



264 Variations of the Arbitrary Constants in Elliptic Motion. 



the result with (D), having regard to (95), we shall have bv 



Zrdbr+dr.br r \ m . : dR x ' >- . 



2— t- +3affndtdR + 2afndtr--j-> (I), if M is not sup- 



• l 



6 



2 





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



m f 

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



M 



m m 



disturbing bodies as m', m", &c, then the factors ™> ttf> &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 2 for r' 2 in (64), then taking the hyperbolic 

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



Ibc 2Sr\ 

 tion (since dt = const.) gives dbv = I — — I dv, and by taking 



be 25 r 





the integral Sv=f[- - ~)dv 9 (K) ; which is the formula given 



c r 



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 ( Sc - 2 j7' rSr \ 



By (64) -T- = - which reduces (K) to iv=f\ — ]dv 9 



c 

 j 



(96), since we have c=v / fl (i_ c 2 ), n 2 a 3 = l, dv= ~[ t ndt, (96) 



.11 i t na * dv ( i dv \ , /T v 



is easily changed to ov= , ^J~dt V ^7 r ^ r I ' ^ ' ' 



which appears to us to be a better form for calculating bv than that 

 given by (K). 



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



plane x, y, the last of (5) which is -^ + — + -j- =0, (M), will 

 enable us to find z, and then by (63), since r f =r 9 we shall get s 



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



T 



