333 



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



PO =6/, is ^ + ^S^^ + B + C - 3M), A, B, C being the three principal 



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

 if A'PCullagh'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 ft 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^ /\ + A sin- \, \ being 

 the angle 0P( = £?) makes with the equator ; and so the central at- 



Q 



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



o 



H-T(l — 3 sin^X), where T=_(A— B) : the attraction of the sphe- 

 roid on P perpendicular to PO and urging P towards the equator is 

 also easily shown to become T sin 2\. 



Now that the point P may be at rest, it is necessary that the tan- 

 gential component of the central force acting along PO should be 

 equal to the sum of the tangential components of the centrifugal 

 force (acting on P perpendicular to the earth's axis), and of the force 

 perpendicular to PO ; this condition gives an equation from which 

 Clairaut's theorem follows instanter, due regard being had to the 

 difference of the polar and equatorial gravities as determined by the 



general expression ^+T(1— 3 sin^ X), and the ellipticity of the ex- 

 terior surface being supposed so small that its square and higher 

 powers may be rejected. 



7. "On the Change of Refrangibility of Light." — No. II. By 

 Professor Stokes, M.A., F.R.S. Received June 16, 1853. 



The principal object of this paper is to explain a mode of obser- 

 vation by means of which the author found that he could exhibit, 

 with ordinary day-light, the change of refrangibility produced by 

 substances opake as well as transparent, even when they possessed 



Proceedings of the Royal Society. Vol. VI. No. 98. 23 



