Variations of the Arbitrary Constants in Elliptic Motion. 257 



V M«(l— e^), (65) ; regarding R as a function of x, y, z, and by 



r ■ . dR dR dR 



(63) as a function of r', v, s, we have -,— dx -{- —j- dy + -f^ dz = 



dR dR dR 



dF ~^ -j^c?y+ "^^*j which by substituting the values of dx, dy, 



,^ , , [ dR dR dR\ dr' I dR 



dz from (63), becomes [x-^-^ +yTy"^'l^) ^ r' + l^^ "^ 



dR\ dR dR dR dR 



'^jdv+ r' -^- ds= ~dr'^^'^ Hv^^^ Is ^*' ^^"^^ "^"^* ^^ ^" 

 .^ . , dR dR dR dR dR rfR 



identical equation, .'.o;^ -^Vlfy +'^ dz =''d?' ""ITy ^^ dx=- 

 dR dR dR 



, J., 



= :/7'(66). 



dv dz ds 



c" c' 



Since /?= — j g'= -j we have by (8), dividing by r' , and substi- 

 tuting from (63 , ^^g- sin. ■»— p cos. v, (67), .*. regarding R as a 



function of 5, and by (67) as a function ofp and 9, we have -y ds=- 



dR . dR dR 



-J- (sin. vdq— cos. vdp)=-ir dp+ -j- dq, which must be an iden- 



dR . dR dR dR 



tical equation, .'. -J- sin. 1;= -7—' -t'cos.v= — ~t-^ or by {66), 



dR dR dR dR 



y~dz'"'d^' '^ii""~ip' ^^^' ' ^^"^® ^"^ ^^ ^^^)' ^^^ ^^^* °^ 



£?R , , c/R c?^ dR dt 



(16), and (17), become dc= — 'i' dt, dp= — -r— — ? dq=-\-' — j 



(69) ; by neglecting quantities of the order of the square of the dis- 

 turbing force, we may evidently write r for r' in all the above equa- 

 tions. 



Assuming M for the unit of masses, (46) becomes n'^a^ — l, (70), 

 also c= \^a{l-e^), (71) ; substituting from these equations in (69), 



•11 1 /—r ^N d^ ^^^^ 



they will be changed to d v a(l — e^) = — -,- dt, dp=^— . 



<ZR andi SR 



X 1—' ^9'=77t^=^X T~' (^2), by the first of these, we have 



cie= V t—1- ^dt-c{l-e^)xd±, which by (39) and (70) 

 Vol. XXX.— No. 2. 33 



