286 
PEOFESSOK CAYLEY ON CUBIC SUEFACES. 
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 (#, w) ; viz. the equation thus becomes 
6 4 x 4 w 3 — 1 2 8xhv s + 6 4 w 7 
+z 2 (-lQxW + U4:XW 4 ) 
4 -zy( 160.TV+ 9 6w 5 ) 
+y 2 ( 48rV4- 80xiv 4 ) 
-f z\ — 27iv 3 
4 -z a y. — 36'xiv 2 
4- z 2 y*. -8x 2 w-{-30w 3 
4 -zy 3 . Mxw 2 
4 -y 4 - 3%x 2 w-\-w 3 
4 -zy. -w 
4 -zy. —x 
■f-zy 5 . w 
+y 6 -x = 0 . 
And if for a moment we write z=a~j- y, y—u. — y and collect, ultimately replacing a, y by 
their values \(z-\-y), \(z — y), the equation can be expressed in the form 
Qiw 3 (x 2 —w 2 ) 2 
4 - 8w 2 (z -\-y) 2 (x +wf(x-\-3w) 
4- 8 w 2 (z —y )\x —w ) 2 (x— 3 w ) 
— 32 iv 2 (z 2 —y 2 ) (x 2 — w 2 )x 
+ l w 0 +y ) 4 0 +w) 2 
— +y T(z-y)(x+w){ 3a?4-7w) 
4 - ±w(z 2 -y 2 )\Ux 2 -27w 2 ) 
— w (z -\-y){z—y) 3 (x—w)(3x—lw) 
t >(z -y)\x-wf 
~ y 3 (z 2 -y 2 )(zw+xy)=0, 
and observing that we have 
zw-\-xy= -z{x—w)-\-x(z-\-y) 
= z{cc+w)-x(z-y), 
we see that every term of the equation is at least of the second order in z-\-y and x — w 
conjointly; and also at least of the second order in z—y and x-\ -w conjointly; that is, 
the surface has the nodal lines (z-\-y=3, x— w—0) and [z—y= 0, #4-w=0), which are 
the reciprocals of the lines 12' and 13' respectively. The nodal curve is made up of 
