PKOEESSOB CAYLEY ON CUBIC SURFACES. 
287 
these two lines and of the line #=0, w= 0 (reciprocal of edge), as will presently appear ; 
so that we have V— 3. 
112. The equations of the cuspidal curve are 
L 2 -12w 2 M=0, 
LM+ 9w 2 N = 0, 
4M 2 + 3LN =0. 
Attending to the two equations 
L 2 — 1 2 w 2 M = y 4 + 8 y 2 xw + 1 Qx 2 w 2 + 2 iyz w 2 + 4 8 w i = 0 , 
LM + 9 w 2 N = y 3 z + 2 y 2 w 2 + 4 xyzvo -{-(8 — 72 = ) — G 4 xw 3 + 1 8iV = 0 , 
these surfaces are each of the order 4, and the order of their intersection is =16. 
But the two surfaces contain in common the line (y= 0, w=0) 7 times; in fact on the 
first surface this is a cusp-nodal line 4#w+ + Ay? = 0 ; and on the second surface it is 
a nodal line w{ixy -\-\^>zw)—^ ; the sheet w — 0 is more accurately 4,ra;-}-?/ 2 -j-By 3 . . . = 0 ; 
whence in the intersection with the first surface the line counts 6 times in respect of the 
first sheet and 2 times in respect of the second sheet; together (5+2 = )7 times, and the 
residual curve is of the order (16 — 7 = )9. 
113. I say that the cuspidal curve is made up of this curve of the 9th order, and of 
the line y= 0, w = 0 (reciprocal of the edge) once; so thatc'=10. In fact, considering 
the line in question y=0, w— 0 in relation to the surface, the equation of the surface 
(attending only to the lowest terms in y, w ) may be written 
xz\y 2 + 4 xw) 2 + w{ — y 3 z 2 ) + w 2 { — 3 Qxyz 3 ) + &c. = 0 , 
4:XW-\-y 2 =Ay%, 
-xz 2 A 2 +^(!-ft)=0, 
that is, A 2 = — 2^g or 4:XW-\-y 2 =\/ — 2 .^yi ; wherefore the line is a cusp-nodal line, 
giving in the vicinity of the line 
and then 
counting once as a nodal and once as a cuspidal line ; and so giving the foregoing results 
6'=3, c'=10. 
114. I revert to the equation which exhibits the 'nodal lines ( x — w=0, y-\-z = 0), 
(x+w= 0, y—z= 0) for the purpose of showing that they have respectively no pinch- 
points; that is, that in regard to each of them we have j'= 0. In fact for the first of 
these lines, neglecting the terms which contain x — w, y-\~ z conjointly in an order above 
the second, the equation may be written 
64 w\x-\-w) 2 (x—w) 2 
+ 8 w 2 (x + w) 2 {x + 3 w) ( z-\-y ) 2 
+ 8 iv 2 {z —y ) 2 (x— 3 w)(x—wf 
