350 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
(A - a) 2 - a 2 # = - x {(7 (7 - a) - 2A] [a (7 - a - 3A) + A}. 
We have 
A 2 — IPS — {(A — a,) 2 — a 2 $} 3 , 
B = ^ (y - a) {S(A- a) 2 + a 2 $} ; 
hence the rational part in question is 
= * fev £Ì Sg ( i'.y+aS +g) K A - a > S + a «)*> 
which putting therein A = 0 gives the value of — yab 2 y- ; and putting A = y, 
gives the value of — ¡3ya~x 2 or — a/3c 2 z 2 . 
57. We have 
1 -S = 
cr (3a — 2) 
3(7 2 — 2(7 — 3 I (7 2 — 2(7 — 
47a 
(y-ay 
3 + s = 
M, 
cr (3(7 — 2) 
3 
(7 (3(7 — 2) 
12 
3(7 2 — 2(7 + •< (7 2 — 2(7 
4ay 
(7 — a ) 2 J _ 
(7 + 
cr (3(7 — 2) V 7 — a 
Hence we have at once the value of 
7 — a 
a = 2 (fi ~ y) <*■ 
where 
08. Moreover 
(A - a) 2 - a 2 $ = A 2 - 2aA + a 2 (1 - 8) 
3(7-2 
[(3cr — 2) {— (7 — a) Act + A 2 } + (7 — a) 2 cr 2 + 30(7] 
where the term in [ 1 is 
(7 2 (7 — a) (7 — a — 3A) + cr {3A 2 + 2 (7 — a) A + 30C7} — 2A 2 , 
and since A = 7 or — a, that is, A 2 — (7 — a) A — <*7 = 0, the coefficient of a- is 
= A {6A — (7 — a)}, 
or the term is the product of two linear functions of a; and we have 
or A = — a,