94 
xMR. J. H. JEAXS OX THE EQUILIBRIUM 
AA’hei’e X is unknown. We therefore have (equation 103) 
4 ^ (3t — - 3 -) + terms in 77 -^, 77 "®, &c. . (118). 
Using the value of So given by equation (ill), we find as the values of 
and cIq (p. 91), 
,7 _ 45 . ^ _ _ 9.65. ,7 _ 6 9 8 7 5 
*^4 — 1> "2 — 14) ^0 — 504 • 
§ 24. Collecting the values of the various constants, we find as the equation to 
the surface (equation 70), 
^77 = 1 +1 (^^ + r) + {(f’ + 77^^) - (f + 77)} (d d- \e^) 
+ [¥ (P + 77^) - ¥4^ (P + r) + 
_|_ 0Z ;2^5. ^^5 _p _ iUoiA j terms in 6'% &c. . (119). 
Tlie occurrence of the indeterminate quantity X can easily be accounted for. For 
if we liave a solution 
¥ = + Gfi + o\f, + e%+ .( 1 20 ), 
corresponding to a parameter 6 Avhich is connected AAuth the rotation liy the relation 
] - orl'27rp = So + S.d- + Sg^^ +.( 121 ), 
then we can obtain this same solution in another form by replacing the parameter 6 
by a new parameter 6 + \6^. As far as d® this leaves the relation ( 121 ) between 
and 6 unaltered, whilst tlie equation to the surfaces as far as 6'^ becomes 
^77 = a"- +{9 + Xe ^) J \ + 6 % -h ^¥3 +.(122). 
It accordingly appears that in equation (119) the value of X is entirely at our 
disposal. We shall therefore take X = 0 . 
Investigation of Stahiliti/. 
§ 25. There is a large d priori probability that the linear series we are now 
considering Avill be stable for small values of 6, but it will be well to rigorouslv 
examine the question. 
It apj^ears from Poincare’s researches that the Avhole investigation reduces to 
determining Avhether the angular momentum is a maximum or a minimum at = 0 .* 
We therefore require to calculate the angular momentum as far as 6'^, and the ansAver 
to our problem will depend upon the sign of the term containing 6~. 
As far as 6~, the equation to the surface in polar co-ordinates (r, <^) is 
(equation (119)) 
= ] -j- cos 2(j) + 20 [r"’ cos 2(f) — 4:('r cos ^) 
_p ^2 ^9_u.^.4 4 ^ — 94 AJ .2 QQg 2(f) -}- .i-23). 
Tlie moment of inertia is given Iia" 
I = dr r d(f) 
* H. PoiNCARK, “Snr la Stahilite He rEqnilibre des Figures Pyriformes 
vol. 198, p. .3.83. 
’ ‘Phil. Trans.,' A. 
