Royal Society. 303 



theory of interpositions. Several of the theorems in this memoir 

 have been previously published by the author, but they are here given 

 along with a great deal of new matter in a connected form, and with 

 the demonstrations annexed, for the first time. 



" On Clairaut's Theorem and Subjects connected with it." By 

 Matthew Collins, Esq., B.A., Senior Moderator in Mathematics 

 and Physics of Trin. Coll. Dublin. 



The author begins his investigations by proving the existence of 

 principal axes for any point of a body, which he makes to depend on 

 the existence of principal axes of an auxiliary ellipsoid (Poinsot's 

 central one) having its centre at the given point, and such that any 

 semidiameter of it is reciprocally proportional to the radius of gyra- 

 tion of the body about that semidiameter. 



He afterwards employs another ellipsoid (called M c Cullagh's ellip- 

 soid of inertia) concentric to the former and reciprocal to it, which 

 admirably suits and facilitates the remainder of his investigations, 

 and whose characteristic property is this, that it gives the radius of 

 gyration itself (and not its reciprocal, as in Poinsot's) about any 

 semidiameter of it, the radius of gyration being in fact equal to the 

 portion of that semidiameter between the centre and a tangent plane 

 perpendicular to it. 



He then proves that the attraction of a body of any shape, whose 

 centre of gravity is O and mass is /z, on a very remote point P along 



PO = d, is -^ + ^(A + B + C - 3M), A, B, C being the three principal 



moments of inertia of the body, and M its moment about OP. And 

 if M c Cullagh's ellipsoid of inertia be taken having O its centre, and 

 its principal axes coinciding in direction with the principal axes of 

 the body at O ; and if a tangent plane to this ellipsoid perpendicular 

 to OP at P' touch it in R, it is shown that the component of the at- 

 traction of the body ju on P in a direction perpendicular to OP is 



parallel to RP, and equal to ^ X OP' x P'R. 



Next comes the proposition, " if two confocal ellipsoids attract an 

 external point, their two resultants are coincident in direction and 

 proportional to their masses," the truth of which is very easily in- 

 ferred from Ivory's theorem. This proposition is then employed in 

 proving that the expressions already found for the attractions of a 

 body of any shape on a very remote point hold true likewise for the 

 attractions of an ellipsoid (whether it be homogeneous, or only com- 

 posed of concentric ellipsoidal strata having the same principal axes, 

 and any variable but small excentricities) on any external point, 

 whether near or remote. 



To apply these reasonings to the case of the earth, the ellipsoid is 

 then supposed to become a spheroid, and the attracted point P is sup- 

 posed on its surface ; then C=B and M=B cos 2 \ + A sin 2 \, \ being 

 the angle OP( = e?) makes with the equator; and so the central at- 



o 



traction along PO, viz. Ji-f _(A-f-B + C— 3M), then becomes 

 a 2 2a 4 



