520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
351 
Similarly 
3 (A - a)' 2 + a 2 £ = 3 (A 2 - 2aA) + a 2 (3 + S) 
= 3<r^2 ~~ ^ ^ ~ 0L ) a ^ + A2 I + o- {(7 - «) o’ + a} {(7 - a) a- 7}], 
where the term in [ ] is 
(7 — a) 2 a 3 — (7 — a) (7 — a + 3A) a 2 + |3A 2 + 2 (7 — a) A — «7} cr — 2A 2 , 
in which the coefficient of a is = A {2A + 3 (7 — a)}, and the term is a product of 
three linear functions: hence 
3 (A - a) 2 + a 2 $ = {(7 — a) a - A} {(7 — a) cr — 2A}. 
59. Substituting these values we have the expression 
1 {(7 — a) cr + a] {(7 — a) cr — 7) {(7 — a) cr — 2A] 2 {(7 — a — 3A) a + A] 3 _ 
ilcr + 7« 
{(7 — a) a — A) (3cr — 2) 2 
which writing therein A = 7 gives — /3<ya 2 x 2 , and writing A = — a gives — afic~z 2 ; we have 
above an expression for —7ab-y 2 requiring only a simple reduction, and the final results 
are 
, _ {(7 - a) cr + a] {(7 - a) a - 2y} 2 {(0 - 7) cr + 7} 3 
(ilcr + 7a) (3cr — 2) 2 ’ 
(3ya 2 x 2 = 
— 'y<xb 2 y 2 = 
(cr — 1) cr 2 {(7 — a) 2 cr + 3a7] s 
(ilcr 4- 7a) (3cr — 2) 2 
_ - li 7 T a > °~ ~ yl 1(7 - «) + 2a l 2 {(«- ft) - g i 3 
(ilcr + 7 a) (3cr — 2) 2 
where it is to be observed that, equating the denominator to 0, we have a triple 
root cr = 00 ; to indicate this, we may insert in the denominator the factor (1 — Ocr) 3 . 
60. We see here the meaning of all the factors, viz. 
Planes. 
Evolute nodes 
Umbilicar centres 
æ = 0 y = 0 3 = 0 =o 
a 
(T 
y — a 
cr = 1 
CJ 
^ 1 
i' 
ll 
b 
-ya 
^ 12 
cr= 
y — a 
0 
II 
b 
— 2a 
cr = —— 
y-a 
<r = l 
“T 
P-y 
— 3ya 
""(y-a) 2 
a 
a-/3 
cr = 00 
Outcrops