412] 
A MEMOIR ON CUBIC SURFACES. 
411 
L- — 12zwM = 0. Next as regards the surface LM + 9zwJ\ r = 0 ; the line y = 0, w = 0 is 
a simple line on the surface, the terms of the lowest order being 9zw (— 4d?x 2 ) = 0 ; 
that is, we have w = 0, and for a next approximation w = Ay 3 , viz. L = y 2 , M = — 2dxy, 
N = — 4 d 2 x 2 , and therefore 
— 2dxy 3 + 9z. Ay 3 {— ^d 2 x 2 ) = 0, that is, 
A =- 
1 
l&dxz’ 
or 
w = — y S ’ ^^ ere i s a threefold intersection with one sheet and a simple 
intersection with the other sheet of the surface L 2 — VLzwM — 0. The surfaces intersect, 
as has been mentioned in the conic z = 0, y 2 + 4am = 0 ; 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, c' = 10. 
Section VII = 12 — B 5 . 
Article Nos. 103 to 116. Equation WXZ + Y 2 Z + YX 2 —Z 3 = 0. 
103. The diagram of lines and planes^) is 
Lines. 
h-» (— 1 
OJ t* Oi to H* O 
VII = 12 - B 5 . 
C5| tO 
X 
h- 1 
II 
tO| 
-ql to 
2 x 5 = 10 
h-i 
X 
Oj 
II 
Ox 
1x10 = 10 
01 
00 
12' 
№ 
<D 
fl 
08 
£ 
13' 
1x15 = 15 
• 
Torsal biplane. 
1x20 = 20 
• • 
• 
Ordinary biplane. 
2 x 5 = 10 
4 45 
. 
• 
• 
Planes each containing 
a mere line. 
Mere lines. 
Ray of ordinary 
biplane. 
Ray of torsal bi 
plane. 
Edge. 
i 
1 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. 
52—2