496 Mr. D. Vaugban on Static and Dynamic Stability 



of the central sphere. The disturbing force on the orbit thus 



becomes 



M's Msgcos(a— W) M.s"ecosu . 



__ __ . _ } , , ^5 j 



M' being put for (M + w), r for D, « for — 77—, s' for——-, and 



21A 2 e 2 D J 



s" for — Yfi — "• Accordingly, in the problem of the two bodies, 



the differential equation -^=-| ^ becomes in the present 



case 



d*r a 2 M' Ws , MVe , N Ws"e 



^- = ^-7^-74- + -^- C08(t>-w) + -^-C08». . (26) 



Multiplying by <?r and integrating, 



dr 2 , a 2 2M 2 Ws -... , Ccos{v-w)dr 



+ 2MV'<?fcosiu/r (27) 



In order to effect the integration of the last terms, which are 

 extremely minute, we may substitute the elliptical values of r and 

 dr in them. The last term integrated in this way gives only 

 periodical quantities. But 



Ccosv — Wj Ce sin v{\—e cosy) 3 . . . 



1 3 ffir== — I — ^ — (cosvcos W -f sin t; sin W) <fo, (28) 



_p being the parameter of the orbit. Noav the angle W being 

 invariable, its sine and cosine may be expressed by the constant 

 quantities H and H' ; and the term becomes 



dv. (29) 



TT , jfsinvcosrfl — ecos vfdv TT C sin 2 r(l — ecosz/) 3 



J P J P\ 



It may be readily found that the first integral consists wholly of 

 quantities multiplied by cosines of v and its multiples, and there- 

 ore periodica], while the second is equivalent to 



— ^-4 1 (1 — cos 2v) (1 - <?cos vfdv, .... (30) 



the secular part of which is found to be 



ellvf. , e°-\ cllv . 



-v^ 1+ 2r r - v near] y- • • • <w 



Equation (27) thus becomes 



dr 2 2M « 2 . 2M's MWUv a 



^= — -^-^+373- ^T- + ^, • (32) 



