412] A MEMOIR ON CUBIC SURFACES. 409 
99. The equations of the spinode curve may be written in the simplified form 
XZW + Y*Z + (a, b, c, d\X, F) 3 = 0, 
4Z (a, b, c, d][X, F) 3 + 3 (4ac — 3b 2 , ad, bd, cd, d 2 \X, F) 4 = 0, 
the line X = 0, F= 0 here appearing as a triple line on the second surface; the 
curve is a partial intersection, 3x4 — 3. 
The node (7 2 is a triple point on the curve, the tangents being the nodal rays. 
The node B 3 is a quintuple point, one tangent being X = 0, 3dY+ 4<Z = 0, and the 
other tangents being given by Z = 0, (4ac - 3b 2 , ad, bd, cd, d 2 \X, F) 4 = 0. 
Each of the facultative lines is a double tangent to the curve, or we have /3' = 6. 
Reciprocal Surface. 
100. Comparing the equation of the cubic surface with that for IV = 12 — 2C 3 , 
it appears that the equation of VI = 12 — B 3 — G 3 is obtained by substituting in that 
equation the values 3 = 0, 7=1. But instead of making this substitution in the final 
formula, it is convenient to make it in the binary quartic V, F) 4 , thus in fact 
working out the reciprocal surface by means of the function 
(xX 2 + yX F — iv F 2 ) 2 + 4zivX (a, b, c, dfX, F) 3 , 
the coefficients whereof (multiplying by 6 to avoid fractions) are 
We find 
where 
The equation is 
viz. it is 
6x 2 + 24 aziu, 
3 xy + 18 bziu, 
y 2 — 2xiv + 12 czw, 
— 3 yw + 6 dzw, 
Qw 2 . 
= L 2 — 12zwM, 
— J = L s - ISzvjLM - 54 z 2 w 2 N, 
L = y 2 + 6 (x + 3c^) w, 
M = 2dxy + 6 (2cx — by + 2bdz) w — 4aiv 2 , 
X = — 4d 2 x 2 — 8d (3bx — 2ay + 2adz) w — 12 (3b 2 — 4ac) w 2 . 
IQgV 2 K i2 — ^~ zw My ~ № — \3zwLM — 54 z 2 w 2 X) 2 ) = 0, 
L 2 (LX + M 2 ) - 18zwLMX - 16zwM 2 - 27z 2 w 2 X 2 = 0, 
where however LX + M 2 contains the factor w, = wP suppose; the equation thus is 
L 2 P - ISzLMX- !6zM s -27z 2 wX 2 = 0. 
Write 
A — 4a- + 1 2c£, 
B = 6cx - 3by + 6bdz - 2aw, 
C = 6bdx - 4ady + 4ad 2 z + 3 (3b 2 - 4ac) w, 
c. vt. 52