FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
251) 
dv 
— COS 0 vdvdOdcf) 
= P- 
(v- — fjr)[v- - 
\ -L 
1 — /3 cos 2 (j)\ jl — /3 cos 2 ( 1 ) 
/3 
1-/3 
d' 
/_, 1 + /S A^^ /I + /5 ..oV /I - /3 cos 2(f)d • 
1 - /3 ^“/ V 
1 ^ 
But 
dv 
dr d 6 d(f) 
, cos- /3 cos- j dr T , 1 p 
dv =-— ^ o, aud tliereiore 
siiF/3 ■ 
cos l 3 cos 7 Aj-Fj- / 1 
siir’ /3 sill- 7 AF \Fj' Aj- 
/ 1 . 1 
1 — T’l ( .-o + ^ 
— 7*1 
/ 1 
Ou comparing this with the expression for Poda. we see that 
dv 
dT 
Pq da 1 Ti (^^^3 + j,^-3 6j rp + (t (^^^3 + 
II 
( 5 ). 
Another form, which will be more generally useful, is found by substituting for 
its value ; it is 
dv 
dT d.6 d(f) 47rp \/i 
, cos-^ /3 cos^ 7 
4r"-r-^- 
sill- 7 
sin- 7 
1 
— 2r 
:,) + G 
III order to express this more succinctly let 
5 - (3 cos- 7 
■ 1 
1 
.(?( 
siiF 7 
1 ^ 1 ^ “ 
A,^" 
1 
1 \ 
-//( 
/' 1 
Ff*A,^ lVA,«y 
vW‘A 
1 
1 1 , 
J AF 
< 1 , = « 
TT AF sill- 7 
G cos- /S cos^ 7 r I 
'F = - 
TT Sill- 7 
^ _ 6 cos'*' /S cos* 7 
TT sill' 7 
JI Ai* Ay)_ 
1 : 
]_ 
AF ’ 
1 
1 
1 _ 1 1 , p. 
FfAp Fi*AiV VF^fiAp IVAf* 
- II 
_7 _ 1 
iVAp Fi^A^i 
_1 
AF ■ 
(<^) 
We note that 
Then 
de d4 « \hj ^ ’ 
>' (17 + V - d) • j 
= ■ ■ ■ 
2 L 2 
(G). 
(G). 
