302 
PEOEESSOE CAYLEY ON CUBIC SUEEACES. 
Reciprocal Surface. 
147. The equation is obtained by means of the binary cubic 
(12w 2 , 4Lzw, y 2 -\-ixw, — 12ufJ7i, X) 3 , 
viz. it is 
432w 6 
+ 7 2w 3 z(Lxw +?/) 
— 64wV 
+(4 xw+y 2 f 
— 2 2 (4^w+^ 2 ) 3 =0, 
or, completely developed, it is 
w\ 432 
d -w 4 . 288 xz 
+w 3 . 72y 2 sd-64-.ir } — 64s 3 
+ w 1 . 4 — 1 6x 2 z 2 
-\-w . 12xy 4 — Sxfz 2 
+ 0 ; 
the section by the plane w=Q (reciprocal of B 6 ) is w=0, y = 0 (reciprocal of edge) four 
times, together with w= 0, y 2 — s 2 =Q, reciprocals of the two rays. 
148. The nodal curve is the line ^=0, w = 0 (reciprocal of edge counting as 3 lines); 
b'=o. In fact the form of the surface in the vicinity is given by w= — ^ y c \ 
viz. there are two sheets osculating along the line in question, that is intersecting in 
this line taken 3 times. 
149. For the cuspidal curve we have 
giving 
jj 12w 2 , 4-zw, ?/ 2 +4a’w 
I 4sw, y 2 -\-Aww, —12 w 2 
12xw+Sy 2 — 4s 2 =0, 
3 Gw 3 +4«2+3f 2 2=0; 
or multiplying the first by 3s and subtracting the second, we have 108w 3 -f4s 3 =0. 
Hence the equations are 
s 3 + 27w 3 =0, 
1 2xw + 3y 2 — 4s 2 = 0 , 
viz. the cuspidal curve is made up of three conics lying in planes through the line s=0, 
w= 0 . 
