412] 
A MEMOIR ON CUBIC SURFACES. 
405 
The equations of the spinode curve may be presented in the form 
¡I XX, 
aX 2 + bZ 2 — F 2 , 
aX 3 + bZ 3 
! x + x, 
TF , 
F 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-t®' - (»-*©'• 
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 
/3' = 2.2 + 2.4 + 1, =13. 
Reciprocal Surface. 
91. The equation is found by equating to zero the discriminant of the binary 
quartic 
y 2 X 2 X 2 + 4w (Xx + Zz) XZ (X + Z) + 4w 2 (aX 2 + bZJ) (X + Zf, 
viz. multiplying by 6 to avoid fractions, and calling the function (*]£X, X) 4 , the coeffi 
cients are 
24cm> 2 , 
6w {x + 2 aiu), 
y 2 + 4 (x + z) w + 4 {a -1- b) w 2 , 
6 w (z + 2bw), 
24 bw~ ; 
and then writing 
L = y 2 + 4 (x + z) w + 4 (a -+ b) w 2 , 
M = ‘k{xz+2(bx + az)w], 
N = Waby' 2 — bx 2 — ay 2 , 
we find 
^1 = L 2 — 12w 2 M, 
— J = L 3 — 18w 2 LM— 54 w 4 N, 
and then the equation is 
i_ 4 |(X2 - l2w 2 M) 3 - (L 3 - 18w 2 LM - 54wUY) 2 } = 0, 
viz. it is 
L 3 N + DM 2 - \8w 2 LMN - l6w 2 M 3 - 27w 4 N 2 = 0.