FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
281 
if + 9 _ 1 i,/7 3 /iV J o cos /3 cos 7 
dr d0 d(f) ^ \kj [ sm® l3 ^ 
1 + cos" y8 
3 sin^ /3 
sin® ^ ^ ' 3 sill" p 
— T ■ 
cos^ ^ cos- 7/1 cI>N .3 1 
D sill" /3 
/i_ _ i .4 7) ^ \] 
When we integrate throughout the region R the limits of r are — eS^ — to 
zero. 
Accordingly 
cos cos 7 ^ ^ 7 i,/,t,oo\ , 1 + cos~/3 
.4. = - (Jj f|{ [2 - M^S^) + ^ 
iJ 
eS, + %f‘S‘ 
+ 
2 - AU^ {L^S, - MVS,^) + <1 
Sin® /3 ^ ^ ^ 3 sill" /3 
+ 
COS^ yS cos- 7 
21) siid j3 
■ Jn 4.4 
.{V AiT,= sV A,T,= + 3 A,T,=J 
+ 
Ps _ £?_ I 4 7 ) 
2Asiny8\^V 
The moment of inertia of the ellipsoid J is 
Also 
Aj = I i¥ ( Af = i/V 
^ \kj siid/3 ^ 
di2 .3 sin® /3 
1 + COS^ /S 3 1 + COS" /d 
5 sin^ 13 ^ 3 sin- ^ 
MP 1 sin® (3 
o-., 
2 _ _ _ . , 
A ' Tvrp cos /3 cos 7 2A'|) ' 27rp ‘ cos /3 cos 7 ‘ 
Lastly, to the required order we may put (^/^o)^ equal to unity in the expression 
for Ay. 
Then 
i (4, - So,*=+A,/) 
_lA! _ p./ . 4 n„ 
^ 47) cos /3 cos 7 \yV“ ^ 
(28). 
This completes the expression for the lost energy E of the system, which may now 
be collected from (19), (20), (24), (27), and (28). 
VOL. CO.— 2 \. 2 o 
