430 
A MEMOIR ON CUBIC SURFACES. 
[412 
Reciprocal Surface. 
147. The equation is obtained by means of the binary cubic 
(12w 2 , 4zw, y 2 + 4<ocw, —12w 2 \Z, X) 3 , 
viz. it is 
432w 6 
+ 7 2w 3 z (4xw + y 2 ) 
— 64 iv 3 z s 
+ (4>xw + y 2 ) 3 
— z 2 (4xw 4- y 2 ) 3 = 0, 
or, completely developed, it is 
w 6 .432 
4- uf.288^0 
4- w 3 .72y 2 z + 64x 3 — 64,2 s 
4- w 2 .4<8x 2 y 2 — 1 Qx 2 z 2 
+ w . 12 xy x — Hxy 2 z 2 
+ f (V 2 -z 2 ) = 0 ; 
the section by the plane w = 0 (reciprocal of B 6 ) is w = 0, y = 0 (reciprocal of edge) four 
times, together with w = 0, y 2 — z 2 = 0, reciprocals of the two rays. 
148. The nodal curve is the line y — 0, w = 0 (reciprocal of edge counting 
as three lines); h'= 3. In fact the form of the surface in the vicinity is given by 
w= ~L y2± ± a/ y 3 , viz. there are two sheets osculating along the line in question, 
that is intersecting in this line taken three times. 
149. For the cuspidal curve we have 
, 12iv 2 , 4zw, y 2 + 4txw =0, 
I 4zw, y 2 + 4<xiv, — 12w 2 
giving 
Ylxw + 3 y 2 — 4^ 2 = 0, 
36w s + 4iwxz + y 2 z = 0 ; 
or multiplying the first by 3z and subtracting the second, we have 108w 3 + 4^ 3 = 0. 
Hence the equations are 
z 3 + 27 w 3 = 0, 
\2xiv + 3 y 2 — 4:z 2 = 0, 
viz. the cuspidal curve is made up of three conics lying in planes through the line 
z — 0, iv = 0. 
The curve may be put in evidence by writing the equation of the surface in the 
form 
(3y 2 + 5z 2 +12xw, 24z, 16^3y 2 — 4z 2 + 12iwc, z 3 + 27w 3 ) 2 = 0, 
where 
16 (3y 2 + 5z 2 4- \%xiv) — \4i4iZ 2 =16 (3y 2 — 4^ 2 4- 12xw).