FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 285 
It will be found from Legendre’s tables that for y = 69° 49''0, k = sin 73° 54'’2 
log -4317642, 
log F' = -2047610, 
log F{y) = -2117987, 
log E = 
log E' — 
-0355145 
-1875655 ^ 
log E{y) — 9-9856045 J 
By integration by parts 
n 
2 ?i 
0 
■l{n - 1)1 + K? 
271 - 1 /c2 
T 2 n 
0 
2 Qi - 1) 1 + 
271 - 1 k"~ 
f«2a-2 
^ 0 
(in - 1) " 
2 n-i 
2n-‘i 
{2n - 1 )k'^ 
T 2n— 4 
0 
L 
( 
J 
Now’ write 
( 86 ). 
(37). 
= 1 (1 -b sec^ /3 + sec^ y), H' = ^ (sec^ /3 + sec^ y + sec^ yd sec^ y). 
The values of ^ and y are 64° 23'-712, 69° 49'-0 ; whence log -8679015, 
log H' = 1-4678555. Also we require hereafter log H — 1-7182664 (see § 3). 
By differentiation 
cl Asin^cos^ 2 ?i.cos-/ 8 cos '7 2 ( 2 )z. — 1 )G cos“/Scos ~7 2 ( 2 ft — 2 )77'cos-/3cos -7 2 ft — 
cie Aff‘ , Ap«+3A 
Whence, by integration. 
Aff‘ A 
+ 
Ay«-2 ^ 
Ap»-iA‘ 
no„, = (?n«. - * 
71 
71 
% 
•ft 
sec-yd sec'^y n 2 ,j_ 4 . . (38). 
On wi-iting 4 / — 1 tan y for sin y, we find that exactly the same formula holds 
good for the T’s. 
To apply this to the determination of Ilo, Tq, we note that 
n “_.2 = cos^y F siiBy E, Tig = F' — sin^ y E' . . . (39). 
Also Tll=)^_{F-E), %=^^(F'-E') . . . (40). 
From the formulae given in Cayley’s ‘ Elliptic Integrals ’ it appears that 
nl:=F + 
T° = A' + 
sin 7 
cos yS cos 7 
sin 7 
cos yS cos 7 
[FE (y) - EF (y)] \ 
>■ 
lF'E(y)-F'F{y) + E'F{y)]\ 
(41). 
Now Eo = F, Eq is given in (40), and E^'^ for n = 2 , 3, 4 . . . are then given 
successively by (37). 
