356 ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
69. We have next 
2 dx 
= da 
7~ a + 2 (7 -a) + 3(^-7) 
n 
(7 — a) a + a (7 — a) a — 27 (0 — 7) a + 7 Ocr + 7« 3<7 — 2] ’ 
which, putting —= 5, and therefore —— = 5 — 1, and = C\ i 
r & 7 —a 7 —a P —7 
4 + 3if 6 
is 
= (2(7 
1 2 
H nn + 
o- + 5—l a — 25 cr + (7 (4 + 3iti) <7 + if 3cr — 2) ’ 
and adding the fractions except ^ the numerator is 
a 2 (274/5+365-4) 
+ a {54 (5 2 - 5) if-+ 725 2 - 805 + 8} 
+ 44/- 165 2 +165, 
which, observing that B' 2 — B=\M, is 
= a 2 (2*7 MB + 365 — 4) 
+ a (^-M 2 + 18M-8B + 8), 
and, substituting for M and 5 their values, this is found to be 
_4(27 + a) 3 , 8 (27+a) 3 a 
- ( 7 -ei)'»" <r ' + ( 7 -a)* 
4 (27 + a) 3 / 2a 
(«■+—) 
V 7-®/ 
70. Hence observing that (7 — —= _-—— , the whole coefficient of da 
& 0 — 7 27 + a 
t (27 + a 
(7 “ a ) 3 
is 
4 (27 + a) 3 ( 2a 
1 a 2 H a 
7 — a 
+ 
(3cr — 2) (a + 5 — 1) (a — 25) [(34/ + 4) a + M~\ a + G ’ 
and the numerator of this expressed as a single fraction is 
4(27 + a) 2 
(7-a) 3 
a [a + 
2a 
7 — a 
[(2 7 + a) a-7] 
which is 
+ 3 (3<r — 2) (a 2 — a — — Ba) {(34/ + 4) a + M), 
= 3 (3(7 — 2) (<7 2 — a — jM) {(34/ + 4) a + M} 
+ <7 - 35 (3a - 2) {(34/ + 4) a + M}