FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
9 
giving on integration, 
«/, = * Mr D 4 (iP“) - A D :i (PQ) + 1D 2 K] 
+ i#A + *f(f+ |+|r 
A' 
A dx. 
8. The solution is now complete as far as third order terms, but can be expressed 
in a more convenient form. We have found for the whole value of v 3 , 
v, = w s +/w'„ 
= -i [AD 3 (iF)- 3 D 2 (PQ) + DR] -w' a P 
+*/MtD*«F) -*D»(PQ) +JD*R] + 4 .... (42) 
where Z is formed of terms involving the function A, and has its value given by 
2 = (t“ + v + f 2 ) I A dX—^f g A 
+ «y € 
2A + 2B + 2C /J 
rW 
AdX. 
The term Z in rq gives rise to a term e : yZ in 0, and this in turn leads to a term 
/(X) ejz = - c-yz, 
in the function (see §§ 4, 11 of the previous paper), from which the whole solution is 
derived. Using the value for Z which has just been obtained, we readily find that, in 
x, y , ^ co-ordinates, 
i abc 
=-*,«**, 
A dX f. 
(43) 
We found, however (see footnote to p. 32 of previous paper), that for a given 
potential problem, the value of T* is not unique. If any function <t> gives a solution of 
the potential problem, then it was found that any other function of the form 
* + 5H f J>4. (44) 
will give a solution of the same problem, provided that F is any function of x, y, z, and 
X, which vanishes when X = X 7 (i.e., when x, y , z, and X are connected by relation (2)), 
and u is any function of X whatever. Consistently with these conditions we may 
take 
u — A, F = 1 
s^P 
abc e 3 ' , 
AA 
and the new solution (44) becomes 
$ + Wabce' A £ A d\ |.(45) 
VOL. CGXVII.-A. 
C 
