520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
349 
we have 
y _(K-3) (a- 1) 
So- +1 - K ’ 
t 0 _ 3Kcr (cr — 2) + 12a- 
r + 3= 3cr + 1 — K ’ 
(o- - 1) (3a - 2) K 
A + V ~ Sa+l-K 
or substituting for K its value we have 
y, 1 K(a-2) + 4i 
A + 1 3<r + l—iT ’ 
Y_ 3 (o- - 1 )jA r (a - 1) + I}. 
3 cr + 1 — K 
_ _ 3 (<r — 1) \K(a- 2)+ 4}. 
Sa+l-K 
K (<r - 2) + 4 = - -) 3 io- 2 - 2(7 - 7 -^ 
ya 
(7 ~ a ) 3 
yet 
(7 - a ) : 
(7 + 
2 
,7 
y — ctJ \ y — a 
3cr + 1 — K = {(3cr + 1) 7a + (7 — a) 2 <7}, = (Ho- + 7a), 
if as before O = /3 2 — yet; and the result is 
f = -& 2 -|(7-a)c7 jl - 
3 (cr 
+ 
2a 
2 7 
7 — a/ v 7 — a 
(7 (3(7 — 2) 
' s\r / il(T + ya l v 
2a \/ 27 
7 — a/ V y— cl 
3 <7 + 
(7 (3(7 — 2) 
and changing the sign of the radical we have the values of tj 1 . 
06. Write for a moment 
A - \ (7 - «) o' -1 “ 
3 (7 + — 
2a 
7 — a 
2 7 \ 
/ — a/) 
(7 (3(7 — 2) 
= (A — a + a V S) 3 = ^4 + i? V>S, 
3 (7 + 
2a 
2y 
(7 (3(7 — 2) 
= 4'+.BV£; 
then in the product of these two expressions the rational part is = A A + BB'S; but 
from the manner in which they were arrived at we have 0 = AB +AB, and the 
rational part is thus 
=-