Professor Mac Cullagh's Lectures on the Attraction of Ellipsoids. 391 



case, whatever be the position of the attracted point. In order to show this, 

 we shall first prove it for a homogeneous ellipsoid of small ellipticities. Such 

 an ellipsoid being given, another, confocal with it, can be constructed so 

 small, that the distance to the attracted point may be regarded as very great, 

 compared with the axes of this ellipsoid : the components of the attraction of 

 this small ellipsoid on the distant point are given by the expressions (10) and 



obtained, and, perhaps, with greater facility, by introducing the consideration of the etatical mo- 

 ment 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 magnitude would be 



M 

 r"' 

 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 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- I'T, z'X -x'Z; 

 but from (7), 



i/T- y'X = ^— -^ — - cos a' cos /3'= - -^ (a- - b') cos a' cos /3', 



3 (B - O ZM 



y'Z - z'Y = - cos §1 cos 7' = — —5- (6- - <?) cos /3' cos 7', 



<^r ^-7 3(C-4) , , 33f,. ,, 



tX - xfZ = ^—^5 — - cos 7' cos a' = - -— (c? - a') cos 7' cos o'. 



Now it is well known, that ^ (a- - 6-) cos a! cos /3', 4 (4" - c') cos /3' cos 7', t (c' - cC) cos 7' cos a', 



are the areas of the projections of the triangle OST on the principal planes. Hence it follows, 



that the resultant moment lies in the plane of the radius vector OT, and the perpendicular OS to 



a tangent plane of the ellipsoid of gyration; the tangent plane being perpendicular to OM. It 



appears also, that the magnitude of the resultant moment is 



^ M 

 -y^(OSxST), 



and therefore that the transverse component of the attraction 



p = _ il^ (OS X ST). 



Or the values of the central force and the moment of the transverse force may be obtained 

 directly from the expression (6) for the potential V. This function is of such a nature, that its 

 differential coeiEcient with relation to any line (the sign being changed) is equal to the re- 



• See Rev. It. Townsend, in the University Examination Papers, 1849, p. 51. 



