PROFESSOR CAYLEY ON CUBIC SURFACES. 
25S 
1, 2, 3, 4, 5, 6 
1', 2', 3', 4', 5', 6' 
12, 13, .... 56, 
where 12 is now the line joining the points 12' and 1'2 ; and so for. the other lines. 
The lines 12, 34, 56 meet in a point 12.34.56; the 30 points 12', 1'2. . . 56', 5'6, and 
the fifteen points 12.34.56 make up the 45 points t ' . 
The above equation, S 3 — T 2 =0, shows that the cuspidal curve is a complete intersec- 
tion 6x4; c'=24. 
Section 11=12 -C 2 . 
Equation W («, b , c, f, g, 7i]£X, Y, Z) 3 -j-2/cXYZ=Q. Article Nos. 47 to 59. 
47. It may be remarked that the system of lines and planes is at once deduced from 
that belonging to 1=12, by supposing that in the double-sixer the corresponding lines 
1 and 1', &c. severally coincide ; the line 12, instead of being given as the intersection 
of the planes 12', 1'2, is given as the third line in the plane 12, which in fact represents 
the coincident planes 12' and 1'2. 
