PROFESSOR CAYLEY ON CTJBIC SURFACES. 
283 
giving the two sheets 
{(4x-\-12cz) 2 — 144cxz— 144bdz 2 }w — 24dxzy=0 and w= 0; 
in regard to the last-mentioned sheet the form in the vicinity thereof is given by w=Ay 3 , 
viz. we have approximately L=?/ 2 , M=2 dxy, and thence if—12z.Ay 3 . 2dxy=0, that is, 
A — o^dxz or w= 24 \xzy Z ’ ^he ^ ne * s thus a flecnodal line on the surface L 2 — 12;swM=0. 
Next as regards the surface LM+92;wN=0 ; the line y=. 0, w = 0 is a simple line on the 
surface, the terms of the lowest order being 9zw(— 4d 2 x 2 ) = 0 ; that is, we have w=0, and 
for a next approximation w=Ay 3 , viz. L=y 2 , M= — 2 dxy, N= — 4d 2 x 2 , and therefore 
— 2dxy 3 +9z . Ay 3 ( — 4 d 2 x 2 ) = 0, that is, A= — 18 ^ — , orw=- there is thus a three- 
fold intersection with one sheet and a simple intersection with the other sheet of the 
surface L 2 — 12;swM=0. The surfaces intersect, as has been mentioned in the conic 
2=0, y 2 -\~4xw = 9 or we have the line y= 0, w = 0 four times, the conic once, and a 
residual cuspidal curve of the order 4 . 4 — 4 — 2, =10 ; that is, <?'=10. 
Section VII=12-B S . 
Equation WXZ+Y 2 Z+YX 2 -Z 3 =0. Article Nos. 103 to 116. 
103. The diagram of lines and planes* is 
Lines. 
Gi to CO tG 
YII= 
12-B 5 . 
o*l to 
X 
X 
X 
X 
IT 
Zjj 
1^ 
tn 
L 
j? 
o 
S 
01 
1X15 = 15 
. 
| : 
Torsal biplane. 
00 
. . 
• 
1X20= 5 
. 
Ordinary biplane. 
Plane: 
• 
• 
13’ 
o 
7 \ 
X 
• 
Planes each containing 
a mere line. 
34 45 
Mere lines. 
j Eay of ordinary 
biplane. 
Eay of torsal bi- 
plane. 
Edge. 
* The marginal symbols in the preceding- diagrams constitute a real notation of the lines and planes ; but 
here, and still_more so in some of the following diagrams, they are mere marks of reference, showing which are 
the lines and planes to which the several equations respectively belong. 
2 Q 2 
