FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
279 
Since on integratioii the terms involving products of unlike harmonics will dis¬ 
appear, we have, as far as material. 
dV 
dn 
, w r „ M . “ / p: 
— =: I ^ X ' - 
S \3 
Now 
\pVq do- = 3 -Ldf — ^d6il(f} 
h \3 
Since the term which is being determined is of the fourth order in e, we may put 
h/lc^^ = 1, and we have 
4/5 )£ 8 I « 
0 0 
.s \3 
L 
(27). 
'^0 0 
Since i9o(^) — ^ == term in S corresponding to f = 0 
vanishes. 
§ 13. Terms in the Energy Depending on the Moment of Inertia. 
We have to determine N,., the moment of inertia of the region R considered as 
filled with positive density. 
In order to obtain this result, we must express if -\- in terms of surface 
harmonics. This was done in § 12 of “Harmonics,” but as a different definition of 
and SI was adopted there from that which I shall use here, it is easier to proceed 
ah initio. 
Let zz: I — and 
{rY - i(i + d), =. Ki + + m- 
For both tlie suffixes 0 and 2, we have (f + — 1, and 
^ - 2 
/<- 
0 0,^1 192 ^.2 '2 
'-'^1 /.l /-I, 1 o 0, , /o /-I _ ? 7 
-- K-' = q" 7 -, k' ~ <r = 0 — k “ - 
l-2f 
1 - 2ff 
In accordance with ecjuation (10) of “The Pear-shaped Figure” I define the 
harmonics as follows :— 
S.2 = (k’ siiff ^ ~ (Jq^) (pif — cos^(^), 
SI = {I siiF 6 - ql) {q'l ~ cos" 4>). 
Now y'' = Jd {id — 1) cos" 9 sin" <^, 2 :" = /dk sin" d (1 — k" cos" f). 
and. 
o 1 — T, . 2t cos^ /3 cos- 7 
”' = ’■1 = A.nV • 
