FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
7 
From §§ 6 and 4, we now have as the value of u 2 . 
u 2 = g 2 +v 1 g 1 
=: —i (APj»+BP/ + CP/) + Q -±P (AP tt + BP„ + CP tt ) 
= — iDP 2 +Q. 
(33) 
The value of w 2 can next be found from equation (26). The right-hand member 
reduces to — V 2 u 2 , and the equation, expressed in £ »/, f co-ordinates, becomes 
. dw 2 3D 
4 aC = -ax 
U 2 
— 1 ^ DP 2 O 
“*3a 
of which the solution is 
Wo = P 4 D 2 P 
|DQ 
and similarly equation (27) leads to 
w ' 2 — — tbV« D P- + Pj- DQ. 
(34) 
(35) 
These values for w 2 and w' 2 are identical with those obtained in the earlier paper, 
although obtained by a slightly different method. We now proceed to the third order 
terms. 
On substituting for u :i its value as obtained in § 6 , we find, in place of equation (28), 
- 4 ( 2 i^ + + + 
which, after a good deal of simplification, 
= V 2 (g, + v,g a ) +V 2 («r,P) +MV*P + 4 P 
dw' 2 
dx 
= V 2 (g,+v,g a )+ 22 ip ( % + («,+/«/,) V«p-4P^.. 
3X 
(36) 
Introducing the various values which have been obtained for g :h g 2 , v 1} w 2 and iv' 2 , 
we find, after simplification, that 
v 2 (g 3 +v^) + 22 A Pf^| + w 2 V ! P 
» 9 3A A 2 3B _ A T, 3A A -n 3C 
= pk ' 3x +?A Tx + , ' AB aA +sAB a); + - 
. 3A , 3B , 30 3A 
+ ®A_ + p A Tx + aA—+...+«—+. 
( 37 ) 
