278 
PROFESSOR CATLET ON CUBIC SURFACES. 
92. This, completely developed, is 
6 4 w 6 . ab{a-\- b) 2 { (a -f b)y 2 — (x — z ) 2 } 
-f32w 5 .2«.5J 3(a+b)[(a-2b)x+(-2a+b)z~]y 2 ^ 
L 4 -' — ^) , 2 [>( — 3a-\-ob)x-\-(5a — 2>b)z^ J 
-j-16w; 4 r 3 ab(a 2 — 7 "j 
|H-[5(9 « 2 + 26a5-5> 2 -26a5(«+%2+«(-« 2 + 26^ + 9J 2 > 2 > 2 [ 
( -f (#— z) 2 [&( — 1 2 <z + 5 )# 2 -j- 22abxz-\-a{a— 12b)z 2 ] j 
-f 8 iv 3 f oab\2a— b)x-\-{— a J r 2b)z\y i 1 
J -f \b(-2a+5b)x 3 +b(3a-2b)x 2 z+a(-2a+3b)xz 2 +a(5a-2b)z 3 ]tf l 
[ + 2{x — z ) 2 [ — 2 bx 3 + bx 2 z + axz 2 — 2 ay z ~\ J 
+ 4w 2 f 3 ab{a-\-b)y 6 
J -|-[ 5 ( 9 a — 2 b)x 2 + 8 abxz + a( — 2a + 9 £} z 2 ]?/ 4 
-f 2 [ — Qbx* + bx 3 z — {a-\-b)x 2 z 2 -f axz 3 — Qaz Ar ]if 
+ 4 x 2 z\x-z) 2 
+ 2iv f 2 ab(x+z)y 6 1 
J — \filx z +2bx 2 z+2axz+Zaz*]f L 1 
1 - 1-4 x 2 z\x+z)y 2 J 
+ y\ay 2 -;f){cy 2 -z 2 )=^ 0 , 
where we see that the section by the plane w~ 0 (reciprocal of B 4 ) is made up of the 
line w—0, y=0 (reciprocal of the edge) four times, and of the lines w= 0 , ay 2 — a?— 0 ; 
W — 0 , by 2 —z 2 = 0 (reciprocals of the rays) each once. 
93. The surface contains the line y= 0, w = 0 (reciprocal of the edge); and if we 
attend only to the terms of the lowest order in y , w, viz. 
x 2 z 2 { 1 Q(x — z) 2 w 2 -\- 8 (x-\-z)y 2 w + y * } , 
which terms equated to zero give 
we see that the line in question ( 3 /= 0 , w= 0 ) is a tacnodal line on the surface, the 
tacnodal plane being w= 0 , a fixed plane for all points of the line', it has already been 
seen that this plane meets the surface in the line taken 4 times ; every other plane 
through the line meets the surface in the line taken twice. We have in what precedes 
the a posteriori proof that in the cubic surface the edge is a facultative line to be 
counted twice. 
94. Cuspidal curve. The equation of the surface may be ’written 
(L 2 -12w 2 M)(4M 2 -f3LN)-(LM+9w 2 N) 2 =0, 
