PROFESSOR CAYLEY ON CUBIC SURFACES. 
281 
Y=0 (the axis) three times, and of a curve of the ninth order, which is the spinode 
curve; <f=9. 
99. The equations of the spinode curve may be written in the simplified form 
XZW+Y 2 Z+(a, b, c, dyx, Y) 3 =0, 
4Z(«, b, c, dyX, Y) 3 +3(4«c-36 2 , ad, bd, cd, d 2 JX, Y) 4 =0, 
the line X=0, Y=0 here appearing as a triple line on the second surface ; the curve is 
a partial intersection, 3x4 — 3. 
The node C 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, 3^Y+4Z=0, and the 
other tangents being given by Z=0, (4 ac — 3J 2 , ad, bd, cd, d 2 fX, Y) 4 =0. 
Each of the facultative lines is a double tangent to the curve, or we have j3'= 6. 
Reciprocal Surface. 
100. Comparing the equation of the cubic surface with that for IV=12 — 2C 2 , it 
appears that the equation of VI = 12 — B 3 — C 2 is obtained by substituting in that equa- 
tion the values c5=0, y=l. But instead of making this substitution in the final formula, 
it is convenient to make it in the binary quartic (*JX, Y) 4 , thus in fact working out 
the reciprocal surface by means of the function 
(xX 2 -fyXY — w Y 2 ) 2 + 4:ZwX(a, b, c, dfX, Y) 3 , 
the coefficients whereof (multiplying by 6 to avoid fractions) are 
§x 2 +24 azw, 
3 xy +18 bzw, 
y 2 -2xw-\-12czw, 
— 3 yw-\- Qdzio, 
* Qiv 2 . 
We find 
|I=L 2 -12 2 wM, 
- J = L 3 - 1 8«wLM - 5 4z V-N, 
where 
L =y 2 +Q{x+?>cz)w, 
M —2 dxy + 6 (2cx— by + 2 bdz)w — Aaw 2 , 
N = — 4<ZV— %d(2>bx— 2ay-{-2adz)w— 12(35 2 — 4ac)w 2 . 
The equation is 
I ^g^{(L 2 -l2^M) 3 -(L 3 -18^LM-54^ 2 N) 2 }=0, 
viz. it is 
L 2 (LN + M 2 ) - 1 8zwLMN - 16^oM 3 - 27zWN 2 = 0, 
where however LN+M 2 contains the factor w, —w P suppose; the equation thus is 
L 2 P - 1 8zLMN - 1 6sM 3 - 2 7 z 2 wW = 0 . 
2 Q 
MDCCCLXIX. 
