OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
oU'J 
Piittiny; 
we find 
and 
'C 
T = 
K~ 
3 + ^ + 2iL 
5(1 -/3) 
i(2 + K-q^(4 
4 — hrf 
T 
1 - 2c/ 
Therefore, with the above alternative form for if, 
y,> (v) and W (■') = - tiiQ - - 
1 ^ 3 ^ (/r) and (/r) = - cos 0 ( 1 — ^ sim 6) , 
{(f)) and {(f)) = sin (f){l — — cos" (f> 
(16). 
The seventh of these harmonics, which is of type OES, stands 1)y itself. We 
had 
This gives in the new notation 
P/{-)- 
PsHi^) ^ 
Su(</>)^ 
CO.S a/t (1 — siir yjr , 
sin^ i/r 
sin 6 cos ^ (1 — K- sin" d)-, 
sin 0 cos (f){ l — K ' cos" 
( 17 )- 
The formulm (6) to (1/) give the fifteen sets of tliree functions constituting the 
fifteen harmonic functions of the first three deu'rees. It Avould be easv, alth<iim'h 
somewhat tedious, to find the coefficient by whicli each function is to 1)e multiplied so 
that its definition may agree with that of the })revious paper. 
§ 3. Ex'presaiom for the Solid Harmoiiics in Rectangidar Co-ordinates. 
The three roots of the original cubic equation were v’-', f, -— ^ , and in the 
new notation the three roots of 
0 ) 
.1 .1 1 , /.) -t I 
•r- I/- d.' 1 . , . 1 - /c - cos- (f) 
T—r + .s-r H-— <^~K~ sm' y tire r , Sill' O, -s- 
* — 1 K~ W — i ft)' /f'Slli'gr K~ 
