mi. j. ii. jp:axs ox the equilibrium 
'.I (I 
Tlius the moineiit of ineitia and the ano’ular velocity both increase as 6~ increases from 
tlie value 6 = 0. The moment of momentum is therefore a minimum for the value 
6 — 0, and this proves the stability of the series of pear-shaped figures. 
Ihe Series of Pear-Shaped Carves. 
§ 26. The equation to the curves of this linear series has already been calculated as 
far as d". In order to obtain a still better idea of the shape of the curves I have carried 
the calculation two deo-rees further. The calculation of these last two degrees is 
extremely heavy, and I have omitted all details in oi'der to save space. The method 
is precisely similar to that which was followed in the calculations of §§20-23. 
It was found that the coefficients multiplying terms in 6^' and 6^ were inconveniently 
large, and to obviate this, the parameter has been changed from 6 to 10® "d. After 
making this change we find, as far as 6'\ for the equation to the surface expressed in 
2:)olar co-ordinates, and for the equation determining o)~, 
r~ = (1 -f -ISOO- + -0206^ +...)— •211^r cos 
-f {-8 - T38ffi - -OGOffi + . . .) r- cos 2cj) 
+ {-063^ - •OOGId'^ - -0031^^ . . .) cos S(f> 
+ (•013tl- + -0008+ . . .) cos ^ 
+ (-0036(93 + •000936'^ . . .) ?'3 cos 5(/) 
-h (-00116^3' _p _ _ ^.G pQg 
+ (•00043d3 -f . . .) r‘ cos iff) -f &c. . . . (128). 
1 — oj'/2ttp = -625 - -0196' — -016^3..^^29). 
§ 27, We must next consider within wTiat limits the calculated terms of equation 
(128) will give a good approximation to tlie complete equation. It is clear that for 
given values of r and 6 the worst aiJjoroximation may be expected when (f) = 0. 
Let us therefore consider the function 6> {r, 6), defined lyv 
cp (r, 0) = (1 + -139^* + -023^^ +...)- -211 ^r 
- (-2 + -138^3 + -069^^ + ... )r- 
+ (-0636' - -0064^3 - -0031^^) v-s 
+ ('OlSffi + •0008ffi- + . . . ) 7*4 + (•00366'3 -p •00093d^) )- 
+ (-0011(9''-p ... ) r'3-p (-00043^3+... )7-7 + &c. . . (130). 
The value of (r, 6) is exj^ressed by a douldy infinite series, of which only a few 
terms are known. When i- = 0, 6 = 0, the value of d> is known to he accurately 
equal to unity. For small values of r and 6, equation (130) will give d) Avith 
considerable accuracy, but for larger yalues of r and 6, the terms calculated will be 
inadecpiate to give a good approximation to the value of d’. What then, we 
