654] 
7 9 
654. 
ON CERTAIN OCTIC SURFACES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877), 
pp. 249—264.] 
I. Consider the torse generated by the tangents of the quadriquadric curve, the 
intersection of the two quadric surfaces 
ax- + b y- + c z- + dw 2 = 0, 
a 'a? + h'y- + cz 2 + d 'w 2 = 0 ; 
then, writing 
be' — b'c = a!, ad' — a'd = f, 
ca' —c'a —U, bd' — b'd = g, 
ab' — a'b = c', cd' — c'd = h\ 
and therefore 
af + b'g' + c'hf = 0, 
the equation of the torse, writing for greater convenience (a, b, c, f g, h) in place of 
(a', b', c', /', g', h'), but understanding these letters as signifying the accented letters 
(a', b', c', /', g', h'), is 
af-y^ + b i g 2 z i x i + c 4 h 2 afy i 
+ arf i x i tu i + b 2 g 4 yhv 4 + tffczho 4 
+ 2b 2 c 2 ghx i y 2 z 2 — 2c 2 f 2 ahx l y 2 w 2 + 2 b 2 f 2 agx^z 2 w 2 
+ 2c 2 a 2 hfy i z 2 x i — 2a 2 g 2 bfy 4 z 2 w 2 + 2c 2 g 2 bhfx?w 2 
+ 2a 2 b 2 fgz 4 x 2 y 2 — 2b 2 h 2 cgz i xhu 2 + 2a 2 h 2 cfz i y 2 w 2 
— 2bcg 2 h 2 w i y 2 z 2 — 2cahf 2 iv i z 2 x 2 — 2dbf 2 g 2 w i x 2 y 2 
+ 2 {bg — ch) (cli — af) {af— bg) xry 2 z 2 w 2 = 0.