520 
520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
339 
42. We have then 
_ — R (a? + 2 A) — (a 2 + A) 3 (a 2 — ¿ 2 - c 2 + A) 
^ — a 2 3y + A (a 4 — 6 2 c 2 ) 
and I assume 
and investigate the value of il. 
We have 
?, m 
— R(a 2 + 2A) — (a 2 + A) 3 (a 2 — b 2 — c 2 + A), = a 4 3y + A©, suppose. 
The equation therefore is 
a 4 /3y + AQ _ 93y 
X. I A /~4 Z.2„2\ ~ To 
— a 2 /3y + A (a 4 — b 2 c 2 ) (3 —y) 3 ’ 
that is, 
93y 3A . . 
or writing ^ y > omitting the factor A, and multiplying by (¡3 — y) 2 , 
this is 
(/3 — y) 2 {© + a 2 (a 4 - 6 2 c 2 )} + 3ÌÌ {— a 2 3y + A (a 4 & 2 c 2 )} = 0, 
in which equation 
© = - 2R - a 6 — (3a 4 + 3a 2 A + A 2 ) (a 2 — b 2 — c 2 + A), 
and thence 
© + a 2 (a 4 — b 2 c 2 ) = same + a 2 (a 4 — b 2 c 2 ), 
= - 3a 6 + 3a 4 (b 2 + c 2 ) - 3a 2 b 2 c 2 
+ A {— 6a 4 + 3a 2 (ò 2 + c 2 )} 
+ A 2 (— 4a 2 4- b 2 + c 2 ) 
-A 3 
= 3a 2 3y + 3a 2 A (¡3 — y) 4- A 2 (/3 — y — 2a 2 ) — A 3 . 
43. Hence, substituting for A its value and multiplying by (/3 — y) 3 , we have 
(/3 — y) 3 {© + a 2 (a 4 — b 2 c 2 )} 
= 3a 2 /3y (/3 — y) 3 — 9a 2 /3y (/3 — y) 3 4- 9/3 2 y 2 (/3 — y — 2a 2 ) (/3 — y) 4- 27/3 3 y 3 , 
which is 
= — 6a 2 /3y (/3 — y) 3 4- 9/3 2 y 2 ()3 — y) 2 — 18a 2 /3 2 y 2 (/3 — y) + 27/3 3 y 3 ; 
viz. this is 
= {- 6a 2 (3 - y) + 9/3y} {(3 - y) 2 + 3/3y} 3y, 
= {— 6a 2 (3 — y) + 93y} (3 2 4 37 + y 2 ) 3y, 
43—2