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1. To find the potential at an internal point P of a thin ellipsoidal shell bounded by similar concentric ellipsoids, the law of force being the inversek th power of the distance. Let (x, y, z ) be the coordinates of P; (l, m, n ) the direction cosines of a radius vector drawn from P to any point Q on the surface; and let PQ =r . Let (a, b, c ) be the semi-axes of the ellipsoids, and (α, β, γ ) their squared reciprocals. Then from the equation of the ellipsoid (αl 2 +βm 2 +γn 2 )r 2 + 2 (αlx +βmy +γnz )r - (1 -αx 2 -βy 2 -γz 2 ) = 0.

XXIII. Theorems on the attraction of ellipsoids for certain laws of force other than the inverse square