Variations of the Arbitrary Constants in Elliptic Motion. 261 



sin.M aV'l-e^. dr Se 



, be, or 5i;=te-|- bu-\- -j-. X 7 =fo-f- 



— ,7— (^n'+^E-M +^A-^— +^fZTJ x^'(9i)' 



dv I ^ dR 



or by (B) we have bv=-b'ui-\- —y: iSaf/ndt dK + ^afndt a -?— — 



, \ dr I rdv 1 \ ^e { dv 



dv I dR\ dr lV\ 



^""^Idt \^^ff^^' dR+2afndta -^] + ^, 



1 \ be 



vn 



+ — -; I X~' (F) ; also by taking the variation of (67), neg- 

 lecting quantities which depend on the square of the disturbing 

 force, we shall have bs = sin. vbq — cos. vbp, (G) ; it may here be 

 observed that the formulae which we have found for or, bv, 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 

 behave they are better for calculating br, bv, than any formulae with 

 which we are acquainted. 



Let r', v', s', denote the radius vector, longitude, and tangent of 

 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, ml ; let a' denote half the greater axis 

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

 motion at the same time, then we shall have as in (46), ^n~a'^ = 

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

 have very nearly M' + m = M'+m'=M^, hence by putting M'=l, 

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

 ,n^«'^ = l very nearly, (92). 



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



r'cos. v' r^sin.t;'^ 

 ues of 0?, y, z, x', y', z', or r cos. v, r sin. v, rs, —. , ■ . l-^ , 



r's' r' 



~r:~-f=ro ^ (4)? and putting for brevity -7 =p, we shall have 



V 1 -j-s - V \-\-s'^ 



wi'[rp cos. (v' — v^-\-zz''\ m' 

 R= ^ -_ 



(p24-z'2)2 [r2-2rpcos.(v'-v)-{-p2-|-(^'-^)^r 



