THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 29 



materials for an answer to the question are to be found only through the third - 

 order terms.* Fortunately the method of the present paper admits of extension 

 to the computation of third-order terms, and so it ought to, and I hope will he quite 

 feasible to decide as to the stability or instability of the pear, a question reserved 

 for a subsequent paper. 



The reader who is interested in the main conclusions of the paper rather than 

 in details of theory, method, or calculations, may care to pass directly to 35. 



GENERAL THEORY OF POTENTIAL OF ELLIPSOIDAL BODIES. 



2. We proceed to develop a method for writing down the potentials of certain 

 homogeneous solids ; in particular of ellipsoids and distorted ellipsoids. We are for 

 the present concerned solely with potential-theory the discussion of rotating liquids 

 does not enter before 19. 



As will soon be evident, the problem in potential theory amounts to the following : 

 to write the equation of the boundary of a homogeneous solid in such a form 

 F (x, y, z) = 0, that the potential at the boundary is of the form F' (x, y, z) = 0, where 

 F' (x, y, z) is a function containing the same algebraic terms as Y (x, y, z), but having 

 in general different coefficients. If this can be done, it only remains to equate 

 F' (x, y, z) + |-to 3 (x 2 +y 2 ) to F (x, y, z}, and we have at once, on equating coefficients, 

 a series of equations which will determine the possible figures of equilibrium for a liquid 

 mass rotating with angular velocity &>. 



3. Let F (x, y, z) = be the boundary of any homogeneous solid of density p. 

 Assuming it to be possible,! let V, be a function of position satisfying V-V ; = 4wp 

 at all points of space and coinciding with the potential of the solid at all points inside 

 the solid, and let V similarly be a function of position satisfying V a V = at all points 

 of space, except possibly the origin or other infinitesimal region inside the solid, and 

 coinciding with the potential of the solid at all points outside the solid. 



Then V f must be equal to V at the boundary of the solid, and we must also have 



cV d 



' = , at the boundary, where -y- denotes differentiation along the normal to the 



surface. 



* Since writing this paper, I have been surprised to find that this conclusion is quite clearly implied 

 in a paper which I published in 1902, "On the Equilibrium of Rotating Liquid Cylinders," 'Phil. 

 Trans.,' A, 200, p. 67. See below, 36. 



t I have not examined in any detail the conditions that this may be possible, because the result of 

 the paper proves that it is possible in the cases which are of importance. Similarly I have not examined 

 in detail the difficulties which might arise at the origin or at infinity, because in the final result they 

 do not arise. We are searching for, and ultimately find, a certain solution of the potential equations, 

 and after the solution has been obtained it is easy to verify directly that it really is a solution, and 

 that it involves no complications either at infinity or at the centre of the solid. 



