342 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
ry 3 
Nodal curve in vicinity of umbilicar centre, a 2 x 2 = — ~ , y— 0, &z 2 = 
47. Write 
£ = — b 2 + q , 7] = — b 2 + r , 
%i = -b 2 + q 1} Vl = -b 2 + r 1 , 
a. 3 
F 
Art. Nos. 47 to 49. 
we have to find the relation between q, q 1} r, r x ; first for q, q 1} the equation of 
correspondence gives 
6P 
+ 3Q (— 2b 2 4- q + qi) 
+ P {6b 4 - 6b 2 (q + q 2 ) + q 2 + 4>qq 1 + q 2 } 
+ 3 {- 2 b 6 + 3 b 4 (q + ft) - b 2 (q 2 + 4 qq, + ft 2 ) + qq 1 (q + ft)} = 0, 
that is, 
3 (q + ft) (36 4 - 2b 2 P + Q) 
+ (<f + 9.P + 9i 2 ) (~ 36 2 + P) 
+ 3gft (q + ft) = 0, 
viz. this is 
~ 3(q + q 1 )ay 
+ (q 2 + 4gft + ft 2 ) (y - a) 
+ (,q + ft) = 0, 
whence approximately q + ft = 0; but it will appear that the value is required to the 
second order; we have therefore 
q + qi = ^^^ 2 + + ft*) 
48. Now the equations 
(a 2 + f) 3 (a 2 + v) = (a 2 + £i) 8 (a 2 + y 1 ), and (c 2 + £) 3 (c 2 + y) = (c 2 + £) 3 (c 2 + rji), 
putting therein for £, y, £, y 1} their values, give the first of them 
log (l + i) + 3 log (l + 2) = log (l + + 3 log (l + £) , 
that is, 
r + 3g — ^ (r 2 + 3 q 2 ) + 2 (r* + 3 q 3 ) =r 1 + 3q 1 -~ (r 2 + 3 ft 2 ) + g - 2 (r 3 + 3 ft 3 ); 
and similarly the second equation 
r + dq + ¿ ^ + + 3cT 2 ^ + Sq ^ = n + Sqi + ¿ ( r * + Sqi ^ + ¿2 ( ri * + Sqi *) ’