406 Prof. Collins on the Attraction of Ellipsoids 



where, as before, u = cos 6. Now to represent this geometrically, 

 let OA" and OC portions of OA, OC be the semiaxes of the focal 

 hjT)erbola whose plane is perpendicular to OB, then OA"^ = a^—b^ 

 and OC'2 = c2-6'2=_(62_c2^^_OB'2. And putting now 

 p^=b^- (i2_c2) cos2 e and p'^ = b'^ + {a^ - b-) cos^ 6, and taking 



the point P on the primary semiaxis OA", so that j^t-tt/ = -t-. So 



that as p<b, P will lie between and A"; then drawing PT 

 touching the hyperbola in T, we find, as before, mutatis mutandis, 

 the whole attraction of the ellipsoid on B 



"^ {a^-b%b^-c^) ^ (T^I^z-ai-cTA") ; 

 P, and T, being now, as before, the ultimate positions of P and 

 T con-esponding to ^ = 0. So that, as before, OP,=rOA", and 



/. P,T;= ~ exactly the same as before. 



The whole attraction on A cannot be similarly represented, 

 because there is no real focal conic perpendicular to OA ; but 



ARC 

 the equation — f- t- + — = ^Trp, discovered by the late ingenious 



Professor MacCuUagh, will then serve to find A; where p denotes 

 the density, and A, B, C denote the whole attractions of the ellip- 

 soid on the points A, B, C. 



7. Let a, b, c be the semiaxes of a homogeneous fluid ellipsoid, 

 and A, B, C the attractions on points at the ends of a, h, c, 

 caused partly by the eliipsoid^s own attractions on its parts, and 

 partly by the centrifugal force of revolution about an axis (2c), 

 or by the action of an extraneous force directed towards its centre, 

 and GC distance from the centre, then the ellipsoid will preserve 

 its shape if A« = B6 = Cc. 



For then the whole forces acting on any point xyz of the mass 



A^ B// Cz 



in directions parallel to a, b, c will obviously be — , -y-, and—; 



XV z 



and dividing these by A« = B6 = Cc, they are as -^, •^, and -^-j 



but when the ))oint xyz is on the surface, these last are as the 

 cosines of the angles that the normal at the point xyz makes 



with the axes, as is evident from the equation (— 2" + ^ + "T— ■'^j 



of the tangent plane. Thus the components of the force acting 

 on the point xyz at the surface are as the cosines of the angles 

 that the normal at this point makes with the axis, .•. the direc- 

 tion of the resultant force coincides with the normal or pei-pen- 



