THE FIGURES OF EQUILIBRIUM OF ROTATING LIQUID MASSES. 43 



we have 



f V (X) V (u +fv) d\ = - 4 ( VT (X) v\ , 



Jo 



Inspection of the values obtained shows that the limit of >/r(x)v when X = o is zero, 

 so that equation (51) reduces to V 2 V, = 47r/o, and since V, is equal to the true 

 value of V< at infinity, and is finite at the origin, V' r must be the true value of the 

 internal potential. Thus the potentials are given by, 



V,- = f V W [/+ * (MI +fv l ) + e 2 (u a +fw +/ V)] d\ .... (52) 



Jo 



V,,= f Vr(\)[/+e(?/,+/i' I ) + e s K+>+/V)]rf\ .... (53) 



JA 



in which all the quantities must be transformed into x, y, z co-ordinates before 

 integration. 



When X = 0, u., reduces to \<a ( >/, f) by equation (40), 



i Ix y z \ , -, / , Ix y z\ 



or to -fu> , fa while u^ = x (, *i, <, ) ~ X ( ~> f? ~j ' 

 \a 2 fe 2 (?) fi? u c J 



also Vi, w and w' all vanish when X = 0, so that (cf. equation (50)) 



a 



and the boundary of the distorted ellipsoid is 

 x 2 y 2 z 2 



16. Before proceeding further it will be convenient to examine in detail the 

 first order solutions which can be obtained from the foregoing analysis, classifying 

 them according to the degree n of the algebraic function u lt and, for brevity, omitting 

 the continual multiplier e. 



n = 0. Solution is u^ = K, v t = 0, <f> = K. 



fv\ iyi sy ft t Y*'Z 



n =; 1. Solution is u^ = pg+gi + r^, v l = 0, < *-r- + ^3 + p- 



J\. -D \j 



n = 2. u,= af+/3, ! +vf +2/rf+%ff+27if,, 



+&+ *) 



G 2 



