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}