PROFESSOR CATLET OjST CUBIC SURFACES. 
285 
109. Each of the lines (X- W=0, Y+Z=0) and (X+W=0, Y-Z=0) is a double 
tangent of the spinode octic ; in fact for the first of these lines we have 
160 8 +80 4 + 16S 2 +5 = O, 160 5 +160 3 + 40=O, 
that is, 
(20 2 +l) 2 (40 4 -40 2 +5)=O, 40(20 2 +l) 2 =O, 
so that the line touches at the two points given by 20 2 + l = O ; and similarly the other 
line touches at the two points given by 2d 2 — 1 = 0. 
The edge X=0, Z=0 has apparently a higher contact with the spinode octic, viz. 
the equations X=0, Z=0 are satisfied by 0=0 twice, 0=oo five times; but it must be 
reckoned only as a double tangent. Hence j3'=2 . 2 + 2, =6. 
Reciprocal Surface. 
110. The equation is obtained by equating to zero the discriminant of the binary 
quartic 
X 2 (?/Z — wX) 2 + iwZ 2 (wZ 2 + sZX+aX 2 ), 
viz. calling this (#)£X, Z) 4 , the coefficients (multiplying by 6) are 
(6iv 2 , — 8yw, y 2 -\-lxw, 6zw, 24w 2 ); 
and then writing 
L = 
M = — 2 y 
N = — 4 +-) 6 xw, 
we have 
iI=L 2 -12w 2 M, 
- J = L 3 - 1 8w 2 LM - 5 4w‘N, 
and the equation is, as in former cases, 
L 2 (LN+M 2 ) — 18+ 2 LMN - 16w 2 M 3 - 27w 4 N 2 = 0 ; 
but LN+M 2 and therefore the whole equation divides by w, and we thus obtain 
1 6L 2 ( - wz*+y*x + w(yz + 4a: 2 ) + w 3 ) - 1 8wLMN - 1 6 wM 3 - 2 7w 3 N 2 = 0 ; 
or, completely developed, this is 
w 7 . 64 
+w 5 . 32 ( Syz—lx 2 ) 
+w 4 .16o:( by 2 -\-0z 2 ) 
+w 3 . ( y*-\-S0y 2 z 2 -\-100yzx 2 — 27z 4 + 64o: 4 ) 
+ w 2 . lx(l 1 y 3 z + 1 2 y 2 x 2 — 9 yz 3 — 4 z 2 x 2 ) 
+w . y\ y 3 z-t-12y 2 x 2 —yz 3 -8z 2 x 2 ) 
-r y 3 -z 2 )=0. 
