I. On the analytical theory of the attraction of solids bounded by surfaces of a hypothetical class including the ellipsoid

Abstract

The following investigation is the result of an attempt to simplify the analytical treatment of the problem of the Attraction of Ellipsoids. The application to this particular case, of certain known propositions relating to closed surfaces in general, showed that the principal theorems could easily be deduced without taking account of any other properties of the ellipsoid than those expressed by two differential equations, of which the truth is evident on inspection. In fact if we take the equation x 2 / a 2 + h + y 2 / b 2 + h + z 2 / c 2 + h = k , we see at once that the expression on the left side, considered as a function of x, y, z, h , satisfies the two partial differential equations d 2 u / dx 2 + d 2 u / dy 2 + d 2 u / dz 2 = 2 (1/ a 2 + h + 1/ b 2 + h + 1/ c 2 + h ) ( du/dx ) 2 + ( du/dy ) 2 + ( du/dz ) 2 + 4 du/dh = 0, and these equations express all that we require to know about the ellipsoid, except the fact that the surface is capable of being extended to infinity in every direction by the variation of h , without ceasing to be closed. But it appeared also that the success of the method depended only on the circumstance that the right-hand member of the first equation, and the coefficient of du/dh in the second, are constants independent of k . It was therefore possible to generalize the process by taking indeterminate functions of h for these two constants. As, however, the coefficient of du/dh could always be reduced to a constant independent of h , by taking a function of h as a parameter instead of h , we may suppose, without loss of generality, that this reduction has been effected.