>f 



253 





lying by the action of the disturbing force ; but if the disturbing 

 force = 0, then c, c', &c. are const., or if the disturbing force is 

 very small, they vary very slowly, and it is evident by (26) that the 

 origin of r is always at the focus, and that whether the elements 

 vary or not. In what follows, we shall suppose that the variable 

 section is constantly an ellipse. . 



By substituting the value of s in the first of (26) it is easily chan- 



b* f f 

 ged to r= M + y c M~*cM' (30), put s = the tangent of the angle 



which r makes with the plane of x, y, and v = the angle that the 

 projection of r on the plane x, y makes with the axis of x } v being 





r cos. v 



reckoned in the direction of the motion: then x= , * V 



r sin. v rs /' f 



;?!+^' z= vf^> (31) ; put el =ecos - *» m = - c sin - • 



b 2 



(32) ; by (31) and 32) we easily reduce (30) to r= « Vh 



> 



s 2 



[Vl+s a +ecos. (v - «)], (33). If we suppose that the plane x, 

 y is taken so as to coincide with the plane of the curve in which m 

 is moving, (when its coordinates are x, y, z,) it is evident that the 



b* 



same 



M 



b 



will remain the same, put p'= ir? > (34) ; then since 5=0, (33) be- 



comes r= t-. ? \' (35), which is the well known equa- 



1 + 6COS. [v — a) > n ^ 



tion of the conic sections, the origin of r being at the focus ; hence 



we obtain the same conclusions as by the previous method ; and it 



may be remarked, that (33) agrees with the equation which La 



Place has obtained by a very different method at pa. 155, Vol. I of 



Mec 



0, or the 



elements to be invariable as he has done. Again, if we take the 

 plane of the primitive orbit of m for that of x, y, by neglecting s*, 



h 



tf 



(33) becomes r= i+^T^y W 



(35) 



(36) 



may be considered as belonging to the curve described by m, when 



