Variations of the Arbitrary Constants in Elliptic Motion. 251 



zdx - xdz ydz - zdy cdt -c'dt c"dt 



dt = ~ c ' ~~ 5r~ =</ '( 7 )j where -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) by z, y, a;, add the products, we have jzc-y^-f 

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

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

 c'dy + c"dx=:0, (10), this added to (8) gives c(z + dz) - & fy + dy) 

 + &'{x+dx) =0, (11); which with (8) show that by neglecting 

 quantities of the order dcdz, dc'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 y z,) to be produced to cut the plane of a?, y 9 

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

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



O 



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

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

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

 a/=a?cos. 6+y sin. d, y'=y cos. — x sin. 4, z=y'ten. 9> .:.*— y 

 cos. 6 tan. 9— x sin. 6 tan. 9, (12). 



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



9, d'=c%va. 6 tan. 9, .". tan. 0= — > tan. 9= " > ( ld /J P ut 



p=-> 5=-, (14), multiplying (6) by A, and taking the inte- 

 c c 



grals, we find the values of c, &, c" at any time, and thence 6 and 

 9 will be found by (J 3), also p and q become known by (14), they 



dc"-pdc ' dd-qdc 

 also give dp = — — ±—>dq= — - ' (15). 



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



evident that z and ~r will be of the order of the disturbing force ; 



at 



•*. neglecting quantities of the order of the square of the disturbing 



/ dR dR\ dR 



force, we have by (6) dc= \yfa— x d~y~) dt > dc/ =- x ~fc dt > dc 



dR 



del' dc' 



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



ydR . *<?R , 



