FIGURE AND STABILITY OF A LIQUID SATELLITE. 
221 
to i Jj; I also put sin x = k sin xfj. At the surface of the ellipsoid we have i// = y, 
X = j3, and since v 0 
the integral sign. 
Then we have 
k sin y sin /3 
= - r — these are the values to be used in |0 4 (^ o ) outside 
£ v-wi-iuv ' * £ sec x *** 
Since 
Since further 
we find 
where 
(v 2 — 1) 1/2 (v 2 — 1 /k 2 ) 1 
P 4 (i/) = (P— 1) 4 +5 (i/ 2 —l)+l = 1 + 5 cot 2 x + ¥ cot 4 x, 
P 4 » = Tf (P-l)[7 (P— 1 ) + 6] = ^(6 cot 2 x +7 cot 4 x ), 
P/ (v) = 105 (P-1) 2 = 105 cot 4 x- 
1P+) = 
(v) = 1 + a cot 2 x + b cot 4 x, ' 
a = 5 (l+f/3 0 ), 
b = (I + 3/3 0 + Ysfiii 2 ). 
(65) 
It must be noted further that when at the surface of the ellipsoid, x = A 
It follows then that (p,) = 1 + a cot 2 /3 + b cot 4 (3 ; and from (64) 
3 m. 
1)^(W = - 
8 k 5 
105X r 5 
1 + iaK 
L ' 11 a 
( 1 — 8 A + -HPA 2 ) (1 + « cot 2 /3 + 6 cot 4 yd). 
It remains to consider the evaluation of ( H 1 , which now assumes the form 
a ‘=“£(i 
+ a cot p + o cot p 
+ a cot 2 x T 5 cot 2 x^ 
sec x dip. 
It would no doubt be possible to split the subject of integration into partial 
fractions, and thus obtain an accurate value as was done in the case of l U 3 , but it does 
not seem worth while to undertake so heavy a task, because a sufficiently exact value 
may be obtained by quadratures. 
The method is employed in § 18, p. 294, in my paper on “ Stability,” and may be 
explained very shortly. 
I divide y into 10 or 12 equal parts—say 10 for brevity—and let 8 = yo7- 
I then compute eleven equidistant values of the subject of integration, say u 0 , u u 
... u m corresponding to xp — 0, 28, 38, ... 108 . As a fact it is unnecessary to compute 
the first four of these, because they are practically zero. 
The equidistant values increase so rapidly that they are very inappropriate for the 
application of the rules of numerical quadratures. Accordingly I take an empirical 
and integrable function, say v, such that v 12 = u 12 and v n = u n , and apply the rules 
