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 *) ’