402 A MEMOIR ON CUBIC SURFACES. [412 
Л, U~i , Fj being, it will be observed, functions of x, y, 8z + yw. The transformed 
equation is 
Л 2 (A 2 /u, — AF X + Ih 2 ) + flziv = 0, 
where the term П may be calculated without difficulty: the first term of this is 
= {y 2 + 4 (8z + 7w) x} 2 .47 2 S 2 [x + fiy — fi 2 (8z + yw)].. [ж + f\y — f 4 2 (Sz + yw)], 
the developed expressions of J (Л 2 /л — A F x + Uf) and of y 2 8 2 into the product of the 
linear factors being in fact each 
= ж 4 .7 2 8 2 + a?y . dy8 + x?y 2 . — ЗС78 + xy 3 . Sby8 + y*. — ay8 
+ [ж 3 (— d 2 — 6cy8) + x 2 y (Scd + 9by8) + xy 2 (— 3bd — 4ayS) + y 3 . ad] (82; + yw) 
+ [ж 2 (9c 2 — 6bd — 2ay) + xy (3ad — 9be) + y 2 .3ac] (8z + yw) 2 
+ [ж (бас — 9b 2 ) + у. 3ah] (8z + yw) 3 
+ a 2 8 4 . (8z + yw) 4 . 
The form puts in evidence the section by the plane w = 0, which is the reciprocal of 
the node D, viz. this is a conic (the reciprocal of the tangent cone) twice, and four 
lines, the reciprocals of the nodal rays, each once. And similarly for the section by 
the plane z — 0. 
83. The nodal curve is made up of the lines which are the reciprocals of the 
six mere lines and the transversal; viz. we have three pairs of lines and a seventh 
line, the lines of each pair intersecting at a point of the seventh line, and these 
three points being the triple points of the nodal curve; t' = 3 as before. 
84. The equations of the cuspidal curve are at once reduced to the form 
A 2 + 24 Uzw + 144 \XjZ 2 w 2 = 0, 
AU + (18F— l2/xA)zw+ 42vz 2 w 2 = 0, 
which are two quartic surfaces having in common the conics z = 0, A = 0, and w = 0, 
A = 0; or we may say that the cuspidal curve is a curve 4.4 —2 —2; that is c' = 12.