390 Mr. George J. Allman's Account of 



Substituting these values of cos o, , cos /3i , cos 7, , and observing that 



cos o' cos ai + cos j3' cos j3, + cos y' cos 71 = 0, 

 we have 



or, 



5? M 



P=-~(OSxST). (11) 



The negative sign indicates* that the force P acts in the direction TS, i. e. 

 from the radius vector towards the perpendicular of the ellipsoid of gyration. 

 If the force P be resolved into three others in the directions of the axes, it is 

 evident from the values given in Proposition V. for X, Y, Z, that these com- 

 ponents are 



-^-7i — ^cosa', -i-^5— '' COS ^', -5^-^ cos 7'. t (12) 



Proposition VII. 



An ellipsoid is composed of ellipsoidal strata of different densities and of variable 

 but small ellipticities ; find the components, central and transverse, of its attrac- 

 tion on an external point. 



The values found in the last Proposition for the components of the attrac- 

 tion of any mass on a very distant point, will be found to hold in the present 



* The direction of the force P, which Professor Mac Collagh determines by the interpre- 

 tation of the negative sign, may be very clearly seen from the following considerations. This 

 force exists in every case where the three principal moments of inertia are not all equal, 

 that is, when the ellipsoid of gyration is not a sphere. The greatest axis of that ellipsoid is 

 manifestly towards that part of the body in which there is a deficiency of attracting matter. If 

 we consider now the position of a perpendicular on a tangent plane of an ellipsoid with relation to 

 the corresponding radius vector, we shaU find that it always lies away from the greatest axis. 

 But the transverse force has been shown to be in the plane of radius vector and perpendicular. 

 Therefore, the direction of the transverse force, being towards the preponderating matter, must 

 be parallel to TS. 



t The results given by Professor Mac Cullagh in Propositions V. and VT. may be otherwise 



