Variations of the Arbitrary Constants in Elliptic Motion, 251 



zdx — xdz ydz — zdy cdt —c'dt c"dt 



dt =^-'^'' ~~di~='"^^'^y^ where -g-' — 3— ' -g- are 



evidently the areas described by r in the instant dt, when ortho- 

 graphically projected on the planes {x, y), {x, z), {y, z) severally. 



Multiply (7) hy z, y, X, add the products, we have zc—yc'-{- 

 xc^'=0, (8), similarly by (6), we get zdc — ydc'-\-xdc"=0, (9), .'. 

 by taking the differential of (8) having regard to (9), we derive cdz — 

 c'dy-\-c"dx=0, (10), this added to (8) gwes c{z-{-dz)-c'{y-\-dy) 

 •^c"{x-\-dx') =0, (11); which with (8) show that by neglecting 

 quantities of the order dcdz, d&dy, dc"dx, m is moving in the same 

 plane when its coordinates are x, y, z, that it is when they become 

 x-{-dx, y-{-dy, z-\-dz, and that the plane passes through the origin 

 of the coordinates. Imagine the plane in which m is moving (when 

 its coordinates are x, y, z,) to be produced to cut the plane of x, y, 

 then will the line of common section be the line of the nodes ; put 

 ^= to the angle which the line of the nodes makes with the axis of x 

 reckoned in the direction of the motion, and 9= to the inclination of 

 the two planes. Let x', y' denote the coordinates of m when refer- 

 red to the line of the nodes, as the axis of the abscissas ; we have 

 x'=xco5.&-\-ysm.^, y' = y cos. ^ — x sm.^, z =y' tan. 9, .'.z=y 

 cos. 6 tan. <j3 — a? sin. 6 tan. (p, (12). 



By comparing the last of (12) with (8), we get &=c cos. & tan. 



<p, c"=c sin. ^ tan. 9, .'. tan. ^= —j tan. 9= '(13); put 



c" c' . . 



p=— 5 5-=— J (14), multiplying (6) by <Zf, and taking the inte- 

 grals, we find the values of c, c', c" at any time, and thence ^ and 

 9 will be found by (13), also p and q become known by (14), they 



dd' — pdc dc' — qdc ^ 



also give dp — ? rfg'= 3 (15). 



Taking the plane of ir, y for that of the primitive orbit of m, it is 



dz , , . . 



evident that z and -rr will be of the order of the disturbing force ; 



.'. neglecting quantities of the order of the square of the disturbing 



/ dR dR\ dR 



force, we have by (6) dc= [y-,— —'X-^jdt, dc'= — x-rdt, dd'=. 



dR dt" dc' 



—y-^dt, (16), also (15) become dp= — 3 dq= —> or by (16), 



ydR xdR 



