EQUILIBRIUM OF ROTATING MASSES OF FLUID. 
397 
The following Tables, computed from (66) and (68), give the values of [k, y] and 
k, i, y 
as far as k — 5, and k — 5, i 
5. 
Table of {k , y}. 
{2,7}= it (1 + /3) 4 -l]- 
{3, y} = i [ (1 + /3) 5 — !]• 
{4,7}= it (1 + /3) 6 - 1]- 
{5,y}=d[(l+/3) 7 -ll- 
Table of \ k, i, y |. 
h=2,i = 2; l[(l + /3) 4 -l]- 
* = 4,i = 2; i 
k = 2,i = 3; i [(1 + /3) 5 — 1]. 
k — 4:,i — 3\ | 
& = 2, i = 4; ■§ [(1 + /3) 6 — 1]. 
k = 4, i = 4 ; •§• 
k = 2,i = 5; f[(l + /3) 7 -ll. 
k = 4, i = 5; | 
k = 3,i = 2- i[(l + /3) 5 -l]. 
k = 5,i = 2; \ 
k = 3,i = 3; f[(l+^(l+|/8)-l]. 
jfe=5,* = 3; f 
k = 3,i = V f [ (1 + /3) 6 (1 + #/3) — 1]. 
k = 5,i = 4; 
k = 3,i = 5; [(l + mi + d/3)-!]• 
k = 5, i = 5; f 
Wi¬ 
ll- 
6. Determination of the Angular Velocity of the System. 
The angular velocity of the system must now be determined in such a way as to 
annul the outstanding potential of the first degree of harmonics. 
Referring to origin o, we have from (35) —- oS dw l directly from the rotation 
potential; the remaining terms are u Y -j- v v since the sectorial harmonic term does 
not contribute anything. 
Thus, taking u 1 from (32), and v 1 from (50), we get for the potential 
„ , 47tM 3 w, f i tt 
+ -sTl 1 ! 1 + %) r H ‘ 
+ To e 
Equating this to zero, 
3 arc 2 
47T 
d = 4 3 {i 4- 
/f/\ 3 i — 00 7* _L- I 
3l a _\ t 1+lp-lff. 
+ Tm e 
cl i = 2 i 
/a\ 2 
6 (~ y + 
e,y ( j + l)^ + 2> ri - lA ; 
1 — 1 
i = 2 
(69) 
