PROFESSOR CAYLEY ON CUBIC SURFACES. 
277 
The equations of the spinode curve may be presented in the form 
XZ, aX 2 +bZ 2 -Y 2 , aX. 3 +b Z 3 
= 0 ; 
X+Z, W , Y 2 
it is a curve 3.4 — 2, the partial intersection of a quartic and a cubic surface which 
touch along a line. 
The binode is on the spinode curve a singular point ; through it we have two branches 
represented in the vicinity thereof by the equations 
(w= -*(*)'■ W=-(4)W) and (w=-i(w)’> W=-(il)W) 
respectively. 
90. The edge counted once is regarded as a double tangent of the spinode curve (I do 
not understand this, there is apparently a higher tangency) ; each of the four mere lines 
is a double tangent; the transversal is a single tangent; hence ft = 2.2-\-2A-\-l, =13. 
Reciprocal Surface. 
91. The equation is found by equating to zero the discriminant of the binary quartic 
?/ 2 X 2 Z 2 + ±w(Xx + Z^)XZ(X + Z) + iw\aX 2 + b7J) (X + Z) 2 , 
viz. multiplying by 6 to avoid fractions, and calling the function (#3£X, Z) 4 , the coeffi- 
cients are 
24 aw 2 , 
Gw(x-{-2atv), 
y 2 -f 4(#+z)w + 4 {a + b)iv 2 , 
Qw(z-\-2bw), 
24 bw 2 ; 
and then writing 
L =tf+4(x+ z)w+4:(a+b)w 2 , 
M= 4(xz -f 2bx + azw), 
X = 1 6 aby 2 — bx 2 — ay 2 , 
we find 
iI=L 2 -12w 2 M, 
— J— L 3 —l 8w 2 LM - 5 4w 4 N, 
and then the equation is 
^- 4 { (L 2 - 1 2 w 2 M) 3 - (L 3 - 1 Sw 2 LM - 54w 4 N) 2 } = 0, 
viz. it is 
L 3 N+L 2 M 2 - 18w 2 LMN - 16w s M 3 - 27w 4 N 2 = 0. 
