384 Mr. George J. Allman's Account of 



For let a point K be assumed on the greater axis OAo of the focal ellipse, such 



that 



DA 



from K let a tangent KQ be drawn to the focal ellipse, and let OP be the per- 

 pendicular let fall from on KQ, then -^ denoting the angle AqOP, 



2 2 



0K-. cos= f = " ~^ {<;- + u- {I- -c^)\. cos^ ^^. 



Moreover, 



0K-. cos= ^|r ^ OF' ^ (a'' - c') cos^ yjr+ib'- c') sin' a^. 



Equating these values, and solving for sin'^ i^, we get 



(a- — c")?«' 



sin- -vi- = —, Ty-T—r, T\- 



C"+ u' {a- — C-) 

 Now 



d . (tan KQ - arc AoQ ) = sin -fd . OK* 



(a' - C-) (b' - c') u'du 



2M 



c ^/\c- + ^l■{a'-c')\ V\c' + u'{b'-c') 



By comparing this expression with (1) given in the last proposition, it appears 

 that the attraction on the point C of the portion of the ellipsoid contained be- 

 tween the two conical surfaces whose semi-angles are 6 and 6 + dO respec- 

 tively, is 



r .. ^""^wiy 2x d . (tan KQ - arc A„Q). 

 {a- — r) {r — c') 



Now in order to obtain the attraction of the whole ellipsoid on the point C, 

 we have to integrate the expression given above between the limits m = and 

 u=l, or OK = OAo and OK = OKj ; from which it appears that its value is 



i-Kfifpabc- rp 

 {a'-c') {b'-c') ' 



It is easy to see that the attraction of the part of the ellipsoid contained within 

 the conical surface, whose semi-angle 6 is equal to cos"' u, is 



* Transactions of the Roj'al Irish Academy, voh xvi. p. 79- Proceedings of the Eoyal Irish 

 Academy, vol. ii. p. 507. 



