268 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
69. In the discussion of the equation 
the two surfaces, thus : 
Cubic surface. 
B s , X=0, Y=0, Z=0 
Biplanes X + Y+ Z=0 
ZX-j-mY +nZ=0, 
intersecting in edge. 
Bays in first biplane, 
X=0, Y+Z=0 ; Y=0, Z+X=0, 
Z=0, X+Y=0; 
rays in second biplane, 
X=0, mY+nZ=0; Y=0, nZ + ZX=0, 
Z=0, ZX+mY=0. 
it is convenient to write down the relations of 
Keciprocal surface. 
Plane w= 0, 
Points in w= 0, viz. 
x=y=z and x : y : z=l : m : n, 
in line (m—n)x-\-(n—l)y-\-(l—m)z— 0, 
that is, Xx-\-[jtjy-\-vz=0, or <7=0. 
Lines in plane w = 0, and through first 
point, viz. 
y~zz= 0, z—x=0, x—y=0; 
lines through second point, viz. 
ny—mz— 0, nz—lx= 0, lx—my= 0. 
70. The equation puts in evidence the section by the plane W— 0, viz. this is the line 
<r=0 (reciprocal of the edge) three times, and the six lines, (reciprocals of the rays) each 
once. Observe that the edge is not a line on the cubic ; but its reciprocal is a line, and 
that an oscular line on the reciprocal surface ; the six lines (reciprocals of the rays) are 
mere scrolar lines on the reciprocal surface ; they pass, three of them, through the point 
x =y=z, and the other three through the point x:y: z=l:m:n; that is, they are six 
tangents of the point-pair (reciprocal of the pair of biplanes) formed by these two points. 
71. I do not attempt to put in evidence the nodal curve on the surface ; by what 
precedes it consists of 9 lines, reciprocals of the mere lines. If we denote by 1, 2, 3 and 
4, 6, 6 the lines which pass through the points x=0, y— 0, z — 0 and through the point 
x : y : z — l : in : n respectively, then these intersect in the nine points 14, 15, 16, 24, 
25, 26, 34, 35, 36 ; and through each of these there passes a nodal line which may be 
represented by the same symbol; that is, we have the nodal lines 14, .... 36. Two 
lines such as 14, 25 meet ; and three lines such as 14, 25, 36 meet in a point ; we have 
thus the six points 14.25. 36 &c. triple points on the nodal curve ; as before, V = 9, t'= 6. 
72. The cuspidal curve is given by the equations 
I &W— 2 Jcwt+a 2 , 24(<£wy+<7\j/), —36(4 ImnJcxyziv—A/ 2, 11 = 0. 
Jew , Jch(?—Zkwt-\-<7\ 2(lcwv-\-vA l i) 
Writing down the two equations, 
{¥nf — 2 Jcwt + <r 2 ) 2 — 24 Jcw(Jcwv + <7^) =0, 
(IcV — 2Iavt-[-<T 2 )(Icivv + c\|/) + 1 8w( hnnk xyzw — \{/ 2 ) = 0, 
these are respectively of the orders 4 and 5 ; but they intersect in the line w = 0, <r=0 
taken four times, or say, the cuspidal curve is a partial intersection 4’5 — 4; c'=16. 
