218 The Configurations of Rotating Liquid Masses [ch. yiii 
It is now clear that for F+£« 2 (¿c 2 + y 2 ) to be constant over the surface 
of the mass, we must have 
V + > 2 (¿c 2 + y 2 ) = — rrrypabc 6 ^ + ~ a ~ 1 + cPoj • • .(200"4) 
at all points of the boundary, 6 being at our disposal. Equating coefficients 
we obtain 
Ja- 
Jjt- 
2 тту p dbc а 2 
to 2 в 
J, 
Б 2тгур abc b 2 
в 
.(200-5), 
which are identical with the equations already obtained in § 190, together with 
№*-**-. 
On inserting the value of <f> in this last equation and equating coefficients 
of a?, xy 2 , xz 2 and x, we obtain 
a a ^ 
0 \ — v — 
a e 
^- e êb‘ 
a ‘ = e ik‘ 
a, = 6 
.( 200 - 6 ). 
a i ' 
If we introduce new quantities c 1} c 2 , c 3 defined by 
Ci 
Г 00 
Jo 
XdX 
XdX 
A A 2 G 
XdX 
AABC 
then equations ( 200 ‘ 6 ) are found to assume the form 
a. , . v /3 7 n c 
2a2 (c 2 +c s ) 26 2 Cs ~ 2c 2 ° 2 — ^ a 
2a 2 3 T 26 2 ' 2 i; 2c 2 1 
fb> c ‘ + h (3c ’+'■>=* £; ) 
3a f® XdX /3 
, + 
XdX 7 
;+ 7 
/' 
J 0 
X,c2A. 
Jo ЬД*В 
i \ 
, 6 
/3 
У .. 
a 2 6 4 
7 
a 2 c 4 ' 
°° dX _ f 
.(200-7), 
Д4 
2a 2 Jo ДЛ 2 26 2 J 0 AAB^2c 2 J 0 Д40 
201. The elimination of a/a 2 , /3/6 2 and 7 /c 2 from the three equations 
(200-7) gives 
a 2 b 2 c 2 + p + p) (0i c 2 + CiC 3 + 3c 2 c 3 ) 
- ~ [ci (b 2 + c 2 ) + c 2 (3a 2 + c 2 ) + c 3 (3a 2 + 6 2 )] + = 0 (201*1).