60 [650 
650. 
ON A QUAETIC SUEFACE WITH TWELVE NODES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877), 
PP . 103-106.] 
* 
Write for shortness 
ct = /3- y, f=a-8, af=p, 
h = 7 “ a > 9 = bg = q, 
c = a — /3, h = y — 8, ch = r ; 
then, 6 being a variable parameter, the surface in question is the envelope of the 
quadric surface 
(a + 6) 2 aghX 2 + (/3 + 0) 2 bh/Y 2 + (7 + 6) 2 cfgZ 2 + (8 + O) 2 abc W 2 = 0 ; 
viz. this is 
2 aghX 2 . XagliX 2 — XctaghX 2 = 0. 
There are no terms in X i , &c. ; the coefficient of Y 2 Z 2 is 
fcfg • bfh + (3 2 bfh. cfg - 2/3 bfh. 7cfg, 
which is 
= bcf 2 gh (/3 — 7) 2 , = a 2 bcf 2 gh, = abcfgh. p. 
Hence the whole equation divides by abcfgh, and throwing out this factor, the result is 
p ( Y 2 Z 2 + X 2 W 2 ) + q (Z 2 X 2 + Y 2 W 2 ) + r (X 2 Y 2 + Z 2 W 2 ) = 0, 
or, observing that p + q + r = 0, this may also be written 
p (YZ + XW) 2 + q (ZX + YW) 2 + r(XY + ZW) 2 = 0,