274 
PROFESSOR CATLET ON CUBIC SURFACES. 
A, U„ Y, being, it will be observed, functions of x, y, lz-\-yw. The transformed equa- 
tion is 
A 2 ( A 2 yj— AY, + Uj) -f- £lzw= 0, 
where the term 12 may be calculated without difficulty : the first term of this is 
= {/+4(^+yw>} 2 . 4yTfx + %— i%dz -j- yw)] . . [tf-f %-f^ + yw)], 
the developed expressions of |(A 2 ^— AY,+Uf) and of y 2 cf into the product of the linear 
factors being in fact each 
=x * . y^+x^y . dy^+x^y 1 . — 3cyh-\-xy 3 . 3 byh-^y 4, . —ayh 
-\-[x z {—d 2 —^cyh)-\-x' 2 y{?>cd-\-%yh)-\-xy\—ohd—iayl)- l r y ?> . ad~](6z-\-yw) 
J r[x 2 {^c l —^hd—2ay)-\-xy{?>ad—%c)- 5 r y' 2 . 3 «c](^ + yw) 2 
-{-[x(Qac— % 2 )+y . 3 <z5](&z + yw) 3 
+ a 2 ^ 4 . (fe -f- y w) 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 
3 = 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 -f 24U2Wd-144 j t£2 2 w 2 =0, 
AU + (18Y -12(jijA)zw-\-'12vzV=0, 
which are two quartic surfaces having in common the conics 2=0, A=0, and iv—0, 
A=0 ; or we may say that the cuspidal curve is a curve 4 . 4 — 2— 2 ; that is c'=12. 
