THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 35 



boundary, but with different coefficients. The method is therefore exactly suited for 

 the determination of figures of equilibrium of rotating fluid (cf. 2). 



As a method of determining potentials, the procedure is indirect in the sense that 

 we cannot pass by any direct series of processes from the equation of the boundary of 

 the solid, as expressed by equation (20), to the general function f (x, y, z, X). We 

 must first search for solutions of equations (18) or (19), and then examine what 

 problem is solved. 



An obvious case to examine first is that in which f is an algebraic function of the 

 second degree. In this case V 2 f is a function of X only, so that the equation for f can 

 be satisfied if the last term in equation (18) or (19) is a function of X only. 



EXAMPLES OF GENERAL THEORY. 

 I. A Sphere. 



9. A quite trivial example may perhaps be taken first, namely that of the sphere 

 x 2 + y s +z 2 = a~. It is seen without trouble that any way of forming the function 

 f (x, y, z, X) will lead to a solution, provided that this function involves x, y, z only 

 through x 2 + y 2 + z 2 , i.e., provided the surfaces are taken to be concentric spheres. 

 For instance, we may take 



f(x, y, z, X) = 

 then equation (18) reduces to 



-4ir/> = I Ct/r (X) d\ + 2 (x'-n) ^ (X'), 



Jo 



of which the solution is found to be \js (x) = 







- 

 (X a) 



The potentials are now given by 



where M is written for *Trpa?, and r 2 = x 2 + y 2 + z 2 (X a) 3 . 



II. An Ellipsoid. 

 10. It is readily seen that a solution of equation (19) can be obtained by taking 



l ..... .. . (27) 



F 2 



