252 Variations of the Arbitrary Constants in Elliptic Motion. 

 The squares of (7) when added, by putting c a + & '* -f d 



* a , 



give r 



dx*+dy*+dz 



dt 



rdr 

 dt 



dR 



b\ (18); also put x -^ + 



y 



dR 

 dy 



+z 



dR 



dz 



rR', multiply (5) by x, y, z, add the products, and 



■ 



we get 



. rf-+!#£t££f + M +rR , _o, (19), by *J*+yJy + 



T 



dt 



zdz 



rdr, and (18), we have 



xd*x+yd s y+zd 2 z 



dT a •' 



d(rdr) 



dt' 



dx*+d y* +dz 



dt 

 Mr-6 a 



rd 2 r fa 



(20) ; hence (19) becomes 



d*r 



dt* + 



dR 



// 



d 2 b* 



r 



+R'=0, (21), putMr-&==*, "^- = MR'+-^ 



(22), 



d* 



s 



. . Ms dR" 



and (21) will be changed to J7J+ - 7 + 



Put 



s 



dR 

 dx 



dR 



// 



X' 



ds 



df dR 



dt dy 



dR 



// 



y 



ds 



ds 



df dR 

 dt dz 



0, (23). 



dR 



it 



Z 



ds 



if" 



dt 



> 



(24), then multiply (23) by x, and the first of (5) by " 

 products, and we get after multiplying by dt, 



xd 3 



sd* 



integral is 



xds — sdx 

 dt 



dt 

 , • a , yds-sdy 



f, similarly — j ( =/', 



s, add the 



df, whose 





zdi 



sdz 



dt 



=/"> 



(25), these give cs~yf- x f, c's=zf-xf", c"s=zf'—yf", (26), 

 by the first two of these s{df -cf")=zjf -yff", which compa- 

 red with the third gives c / // +/c // -c/ / =0, (27) ; we also have by 

 (25) cds=fdy-f'dx i c'ds=fdz-f"dx, d'ds=f'dz-f"dy, (28), 

 then by taking the differentials of (26) having regard to (28), we 

 get sdc = yd f-xdf, sd& = zdf - xdf" , sdc"=zdf'-ydf", (29). 



By the first and second of (26) we havecs — c / «4- 



Cf'-cf 



// 



f 



x=0, 



which reduces by (27) to cz - c'y + c"x = 0, and agrees with (8), as 

 it evidently ought to do. 



Now, since r 2 =x*+y*+z 2 , by restoring the value of* in (26), 

 then substituting the value of z from (8) in the first, the value of 

 y in the second, and the value of x in the third, we shall have three 

 equations of the second order ; the first in terms of x and y, the sec- 

 ond in terms of a; and z, the third in terms of y and z; which show 

 that m is constantly moving in some conic section, whose elements 

 are constantly changing, since c, &, c", /, /', /" are continually va- 



