Variations of the Arbitrary Constants in Elliptic Motion. 253 



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

 force = 0, then c, &, &ic. 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= lvf+yivF"~*lVl' ('^O)? P^t 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 op, y makes with the axis of x, v being 



,.,,.. ~ , r COS. V 



reckoned m the direction of the motion; then a; = —7==? m = 



x^l+s^~ ^ 



r sin. V rs f f 



(32) ; by (3i) and 32) we easily reduce (30) to r= ^"/l+ss .^ 



\y\-\-s^-\-ecos.{v~ ttf)], (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'' 

 curve described by m will remain the same as before, and that ^ 



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



P' 



comes r=j-, 7 x? (35), which is the well known equa- 



i + ecos. (v — wj ^ /' "1 



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 ihe equation which La 



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



the Mec. Cel., when we suppose the disturbing force =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^ , 



which is of the order of the square of the disturbing force, and using 



p', (33) becomes r= ^ , ^^^^ (v -zi) ' ^^^'^' ^^'^^ '^ °^ *^^^ ^^^^ 

 form as (35) ; and it is evident that by neglecting quantities of the 

 order of the square of the disturbing force, all the quantities in (36) 

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



