Variations of the Arbitrary Constants in Elliptic Motion. 261 



sin. u 



•i 



c 



de, or 5v 



+ 



•i 



e 



r 



+ 



dr 

 ndt 



Se 



X 



ae 



V 



•l 



r 



(<5n'+<5E-fo) + 



ndt\ - + 



1 



r 



•i 



e 



he 



<fo-f- 



(91), 



dv 



or by (B) we have <5i>=6tf-f j-g [Zaffndt dR + 2a/ 



dR 



A/l-e 2 . fe 



dr / rdv 



1 



• I 



e 



be 

 *7 



(E) ; h 



1 





VI 



+ 



e 



1 



dv 



<5e 



Baffndt dR + 2<i/ndf 



«m\ 



dr /v^l — e 2 



r 



•r 



e 



X~> (F); also by taking the variation of (67), neg- 



lecting. quantities which depend on the square of the disturbing 

 force, we shall have Ss = sin. v5q — cos. t?5p, (G) ; it may here be 

 observed that the formulae which we have found for <5r, $v, explain 

 in a very simple manner what La Place proves by an elaborate and 

 not very obvious process at pp. 291, 292, 293, Vol. I of the Mec. 

 Cel., we would also remark that they are new to us, and that we 

 believe they are better for calculating 5r, <$r, than any formulae with 

 which we are acquainted. 



Let r 7 , v f , s'j denote the radius vector, longitude, and tai w 

 the latitude of m / above the plane x, y, at the time t, v' being reck- 

 oned from the axis of x in the plane x, y, and in the direction of 

 the motion of the bodies m, m' ; let a' denote half the greater axis 

 of the ellipse in which m 1 is moving at the time t, and f,ndt its mean 

 motion at the same time, then we shall have as in (46), ,n 2 a' 3 

 M'-f-wi' ; if m, m', are very small when compared with M', we shall 



M 



M 



M'. hence bv Duttinsr M 



we shall have by (46), and by what has now been proved, n 2 a 2 



,n*a'* 



1 very nearly, (92). 



In order to apply the above formulae, we must substitute the val- 

 ues of a?, y, z, x', y', z', or r cos. v, r sin. v, rs, 



r 1 cos. v' r 1 sin. v' 



rV 



r 1 



•l 



s 



vT+ 



P, we shall have 



R 



cos 



«)+«'] 



m 



(P 2 + 



3 

 3 



[r 2 -2rpcos. {v f 



+ 



i 



