340 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
and the equation thus is 
{— 2a 2 (/3 — y) + 3/37} (/3 2 + /3y + 7 2 ) /37 + il j— a 2 /3y — C“ 4 ~ b 2 c 2 )^ (/3 — 7) = 0, 
or finally 
O {a 2 (£ - 7) + 3 (a 4 - 6 2 c 2 )} = (- 2a 2 08 - 7) + 3/37) (£ 2 + ^7 + 7 2 )- 
But c 2 = a 2 + /8, b 2 = a 2 — 7, and hence cl* - b 2 c 2 = - a 2 (/3 - 7) +/3y, and therefore 
a 2 (/3 — 7) + 3 (a 4 - 6 2 c 2 ) = — 2a 2 (/8 — 7) + 8/37; 
the equation thus divides by — 2a 2 (/3 — y) + 3/37 an d we have 
£2 = /3 2 +/87 + 7 2 , 
or, as this may also be written, £2 = a 2 -• /3y, = /3 2 —ya, = y 2 — a/3. So that £2 has the 
value originally so denoted, and we have then 
77 — — a 2 4- 
9/3y 
(/3-7) 3 
il. 
44. III. Lastly the equation 0 = (a 2 + £) 3 (a 2 +17) = (a 2 + fx) 3 (a 2 + %) is satisfied if 
a 2 + 77 = 0, a 2 + % = 0: the equations 
(6 2 + |) 3 (6 2 + 77) = (6 2 + £) 3 (& 2 + 771), 
(c 2 + £) 3 (c 2 + 77) = (C 2 + 10 s (c 2 + T7 a ), 
then give 
(6 2 +£) 3 = (& 2 + £) 3 > 
(c 2 + £) 3 = (c 2 + £x) 3 , 
which can be satisfied by £ = £1, leading to | = f 1 = — a 2 , which is the case I., or else by 
b 2 + £ = g) (6 2 + £1), 
c 2 + f = w 2 (c 2 + %i), 
that is, 
£ = ft>6 2 + ft) 2 c 2 , = a> 2 6 2 + ft)C 2 . 
To show that these values satisfy the relation between f, | x , observe that they give 
| = - 6 2 — c 2 , = 6 4 - b 2 c 2 + c 4 , 
whence also 
r+4|fx + |i 2 = 3(6 4 + c 4 ), 
and the relation becomes 
Qa 2 b 2 c 2 — 3 [a 2 (6 2 + c 2 ) + b 2 c 2 ] (6 2 + c 2 ) 
+ [a 2 + (b 2 + c 2 )]. 3 (V + c 4 ) - 3 (6 2 + c 2 ) (6 4 - b 2 c 2 + c 4 ) = 0, 
which is an identity