412] 
A MEMOIR ON CUBIC SURFACES. 
383 
of a double-sixer is a double-sixer. Hence the 27 lines of the reciprocal surface may 
be (and that in 36 different ways) represented by 
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 l = 0, shows that the cuspidal curve is a complete inter 
section 6x4; c' = 24. 
Section II = 12 — C 2 . 
Article Nos. 47 to 59. Equation W (a, b, c, f, g, li\X, Y, Z) 2 + 2JcXYZ = 0. 
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 inter 
section 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.