PHILOSOPHICAL TEANSACTIONS. 
I. On the Analytical Theory of the Attraction of Solids hounded by surfaces of a hypo- 
thetical Class including the Ellipsoid. By W. F. Donkin, M.A., F.R.S., F.B.A.S., 
Savilian Professor of Astronomy in the University of Oxford. 
Eeceived September 2, — Bead December 8, 1859. 
The following investigation is the result of an attempt to simplify the analytical treat- 
ment 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 
+ + c2 + 4 — 
we see at once that the expression on the left side, considered as a function of x. y, h. 
satisfies the two partial differential equations 
(Bu (Fu Pu „/ 11 ^ 1 ^ 
dx^'dy'^'dz^ h' b‘^ -P h' + hj 
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 
du 
equation, and the coefficient of ^ in the second, are constants independent of Jc. It was 
therefore possible to generalize the process by taking indeterminate functions of h for 
dvL 
these two constants. As, however, the coefficient of -tt could always be reduced to a 
MDCCCLX. 
B 
