369 



from P the right line PT, touching the hyperbola in T, and 

 denoting the angle EPT by 9, we have 



COS e = ri!lZ_i cos ; 

 P 



whence it may be shown that if p be the point to which P 

 moves when (j> becomes (p -f- d(f>, the interval P^ multiplied by 

 cos 6 will be proportional to the attraction of the matter dM 

 contained between the surfaces of two right cones having B 

 for their vertex and OB for their common axis, provided^ 

 and (j) + d(j> he the angles which the sides of these cones make 

 with OB. The whole attraction is therefore proportiona. 

 to the difference between the tangent PT and the hyperbolic 

 arc ET, the position of T being that which corresponds to the 

 supposition ^ = 0, or p = c. 



" When the attracted point is at the extremity of the great- 

 est axis of the ellipsoid, we' cannot employ a similar method, 

 because there is no focal curve perpendicular to that axis. 

 But if A, B, c be the attractions at A, B, C respectively, we 

 have the known relation 



A B c ^ 

 - + - + - = 47r, 

 a c 



of which a geometrical proof will be found in the Proceedings, 

 vol. ii. p. 525 ; and from this relation we can find a in terms 

 of B and c. 



" The preceding method of treating the question of the 

 attraction of ellipsoids was given at my lectures in Trinity Col- 

 lege, in the beginning of last year. I have since observed 

 that the same results may be obtained, and perhaps more rea- 

 dily, by dividing the ellipsoid into concentric and similar shells. 

 For the attraction of dM is equal, in each case, to the attrac- 

 tion of the shell bounded by the surfaces of two]_ellipsoids 

 whose semiaxes are acos^, bcoS(p, ccoscp, and aeos{<f) -\- d<j>), 

 6cos(^ 4- d^), ccos(^ -]r dtf), these ellipsoids having O for their 



