THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 33 



G. As a matter of convenience, involving neither loss nor gain of generality, we 

 shall write 



(16) 



in which \[s(\) is any function of X. and f(x, y, 2, X) a quite general function of x,y, z, 

 and X. Then, from equation (ll), we must have 



f(x, V ,z,\) = .......... (17) 



identically at all points, when X has the value appropriate to the point x, ?/, 2. We 

 accordingly have 



so that equations (14) and (15) reduce to 



A' 

 47TH = 



'. (i.) 



Moreover the family of surfaces (X = cons.) may now he supposed to he determined 

 hy equation (17), and the boundary will he given by 



f(x,y,z, 0) = (20) 



Thus, to sum up, if y and i//- are such that either equation (18) or (1'j) is satisfied, 

 then the potential of the homogeneous solid of density /> whose boundary is 

 determined hy equation (20), will be given by 



V f -V u = W= V(x)/(,;', ,j, z', \)d\, (21) 



the value of W being evaluated at the point x', y', z', and X' being determined from 

 the equation f(x', y', z' ', X') = 0. 



7. The boundary X = is of course fixed by the solid whose potential is required, 

 but we are left with a certain amount of choice as to the disposition of the surfaces 



f A 

 and \p is any function of A whatever. Replacing ^ (A') - \j/ (0) by u (A) d\, we find that if <I> is a 



solution then 



will also be a solution of the same problem. 

 VOL. CCXV. A. 



