414 
A MEMOIR ON CUBIC SURFACES. 
[41.2 
111. To transform the equation so as to put in evidence the nodal curve, 
I collect the terms according to their degrees in (y, z) and (x, tv); viz. the equation 
thus becomes 
64 x 4 tv 3 — 128x 2 w 3 + 64w 7 
+ z 2 { — \Qx 3 w 2 4- 144icw 4 ) 
+ zy ( 16(te 2 w 3 4- 96w 5 ) 
4- y- ( 48oc?tv 2 + 80.tw 4 ) 
+ z 4 . — 27 w 3 
4- z 3 y . — 3 dxtv 2 
+ z 2 y 2 . — 8x 2 tv + 30tv 3 
4- zy 3 . 44xw % 
+ y 4 .12 x-w + w 3 
+ z 3 y 3 . — w 
+ z-y 4 . - x 
+ zy 5 . w 
+ y 6 . x = 0 ; 
and if for a moment we write z = a + y, y = a — <y and collect, ultimately replacing 
a, 7 by their values \ (z + y), \ (z — y), the equation can be expressed in the form 
64w 3 (¿c 2 — w-f 
+ 8iv- (z + y y (x + w y (x + 3iv) 
+ 8w 2 (z — y y (x — w y (x — 3w) 
— 32w 2 (z 2 — y 2 ) (x 2 — tv 2 ) x 
+ \w (z +y) 4 (x + w ) 2 
— w (z + y ) 3 (z — y)(x + tv) (3x + 7tv) 
+ (z 2 — y 2 ) 2 (1L» 8 — 27w 2 ) 
— w (z + y ) (z — y) 3 (x — tv) (3x — 7w) 
4- \w (z —y) 4 (x — tv) 2 
— y 3 (z 2 — y 2 ) (ztv 4- xy) = 0, 
and observing that we have 
ztv 4- xy = - z (x — tv) 4- x (z 4- y) 
= z(x + w) -x (z - y), 
we see that every term of the equation is at least of the second order in z 4- y and 
x — w conjointly; and also at least of the second order in z — y and a?4-tv conjointly;