364 On the Attraction of Ellipsoids. 



we have 



P = -- 77- . = 



r 4 jt?r sin r 



or, 



O 1 J" 



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 direction of the axes, it is evident from the values 

 given in Proposition V. for X, T, Z, that these components are 



3 (-4-7) , 3(5-7) a , 3(0-1) 



,i ' COS a', ' COS /3 , , 4 - COS /. t (12) 



* The direction of the force P, which Professor Mac Cullagh determines by the 

 interpretation 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 of the system at are not all equal, that is, when the ellipsoid of gyra- 

 tion 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 now con- 

 sider the position of a perpendicular on a tangent plane of an ellipsoid with rela- 

 tion to the corresponding radius vector, we shall 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 from T to S. 



t The results given by Professor Mac Cullagh in Propositions V. and VI. may 

 be otherwise obtained, and, perhaps, with greater facility, by introducing the con- 

 sideration of the statical moment of the attracting force. * 



If the three principal moments of inertia were equal to each other, then the 

 whole attraction would be in the direction of the centre of gravity, and its mag- 

 nitude would be 



M 



7*' 



In general, however, the attracting mass will be of an irregular shape ; there will 

 exist then, in addition to the principal part of the attraction, which will be central, 

 a transverse force which will tend to cause a motion of rotation about the centre of 

 gravity. 



The components of the moment of this transverse force in the three principal 



planes are 



x Y - y'X, y'Z - 2' Y, z'X - x'Z; 



* See Rev. R. Townsend, in the " Dublin University Examination Papers," 1849, p. 51. 



