FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
17 
Let us limit ourselves to the terms in e\ Using the value of V 6 given by equation 
(56) and that of Id assumed in equation (52), we find that we must have 
- Trpabce’x (i t n x 4 + C 22 y 4 + C 33 z l + C V2 x 2 y 2 + c 31 zV + c 23 ?fz 2 + ti X x 2 + % 2 + tqz 2 + tq) 
= 
— 7 TfJ 
abcQd\ 
X x 4 
Y^a 8 
b s c 8 
2 2 
+ 21^- 
6V 
2 2 
+2m^ 
c a 
+ 4 +] 
■in- 
coefficients we 
obtain 
Cn 
_ 1 
— 4 
4^; 
a 
C 22 
_ 1 
— 4 
4s»; 
a b 
t .33 = 
1 b vr 
4 2 8 ’ • 
ac 
■ ■ ( 73 ) 
U 
_ 1 
— 2 
f) j 
aW 
J Gi 
_ 1 
— 2 
0 
6 4 m ’ 
arc 
Cl2 = 
1 A 
inuU T » • • 
a b 
• • ( 74 ) 
Pi 
_ 1 
— 2 
0 
- P; 
P 2 
_ 1 
— 2 
0 
2L4 4 > 
P 3 = 
1 0 r 
. . ( 75 ) 
a 
a b 
arc 
Pi 
_.i 
— 4 
A. . 
a 
. . (76) 
These equations, in addition to those found for the first- and second-order terms in 
the previous paper, express the condition that the third-order figure (72) shall be a 
possible figure of equilibrium. 
15. On substituting the values of C n , C 12 ... C 33 which have been obtained in § 12 into 
the six equations (73) and (74), we obtain a system of six equations from which it is 
possible to determine the six unknowns 1L, jit, AT, l, Tit, It, The solution is actually 
effected in § 17 below. 
If we substitute the three values of tq, tL and ti 3 obtained in § 12 (equations 
(62)-(64)) into the three equations (75), we obtain three equations which can be 
written in the form :— 
m f7 +* 
3 | Ad\ | f ([/\ 
A A 2 
-if 
J l 
__j i JL 0 i 
0 AAV 4 Uo AA 2 
., i r rcdx 
1 + 4 C 
A A 2 
4-2 r 
* u 
_G 
A A 
2 
Ip 
a d\ 
AAB 
A d\ 
6 
+ 
l 
AAC +4q J 0 A AC 
a 2 ¥ 
’ Bd\ 
,a ° B dx 
0 AAB 
l '( i +i 
arc 4 
dx 
o AAB 2 
C dX 
+|r 
Cdx 
0 AAB 
dx \ 
= A-2 
AAC 2 Jo A AC 2 ) ^ 2 
r 00 / i 
Lt 
Jo A-A.B 
G 
dX , 
AAC 
dX. . 
(77) 
(78) 
(79) 
It will be seen that p, q, r do not occur on the right-hand sides of these 
equations. 
yol. ccxvii.—A. 
D 
