

Variations of the Arbitrary Constants in Elliptic Motion. 257 

 V'M^l — e 2 ), (65) ; regarding R as a function of x, y, z, and by 



• . r dR dR dR 



(63) as a function of r^ u, s, we have ~J~ dx -\- —j- dy -\- -r;dz = 



-• 



-rjar'-j- -j— dfl-{- "j - ds, which by substituting the values of dx, dy, 



t dR dR dR\ dr 1 I dR 

 dz from (63), becomes [x -^ +y ^ +z -jj) X fl + [x-jj -y 



dR\ dR dR ■ dR dR , 



~7Z)dv-\- i* j* ds— j-jdr'-ir -r~dv-{- -r- ds, which must be an 



dR dR dR dR dR dR 



identical equation, .>*^ +y^ +* d ~ =r> d? > *^ -y rfj 



<*R dR dR 



d,'^ = ^'( 66 )' 



Since p= — j gr= -> we have by (8), dividing by r', and substi- 



* 



tuting from (63 , s-=$sin. v — p cos. «, (67), ."• regarding R as a 



dR 



function of s, and by (67) as a function ofp and 5, we have -r- ds 



dR dR 1 dR , 



■j- (sin. udg' — cos. vdp)=~r- dp-\- -r* dq, which must be an iden- 



dR dR dR dR 



tical equation, •". -7- sin.. »== -r-» -7- cos . v = — -r- > or by (66) 



dR _ d_R dR _ dR 

 *Tz ~~dq ' X dz ~~ dp' 



i 



dR _ ' dR dt _ dR dt 



(16), and (17), become dc= - ^ dt, dp= - -^ -, dq= d y 1- 



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

 turbing force, we may evidently write r for r 7 in all the above equa- 



tions. 



M for the unit of masses, (46) 



also c = V 



•a(l-e»)=--j-<Mp 



tions in (69) 

 dR andt 



V\ 



e 



dR andt dR 



X 



VI 



e 



(72) 



ede= v/ 1 " 6 ' ^?d*~c(l-e a )Xd-L, which by (39) and (70) 



v a dv K J 2a J v ' 



Vol. XXX— No. 2. 33 



