A MEMOIR ON CUBIC SURFACES. 
413 
412] 
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 
that is, 
166/8 + 8 6> 4 + 160 2 + 5 = 0, 160 5 + 166» 3 + 40 = 0, 
(26 2 + 1 ) 3 (4<9 4 - 46> 2 + 5) = 0, 46 {2d- + l) 2 = 0, 
so that the line touches at the two points given by 2d 2 +l = 0; and similarly the 
other line touches at the two points given by 29' 1 —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 6 = 0 twice, 6= go five times; but it 
must be reckoned only as a double tangent. Hence yS' = 2.2 + 2, =6. 
110. 
quartic 
Reciprocal Surface. 
The equation is obtained by equating to zero the discriminant of the binary 
X- (yZ — wX)- + 4iuZ- (wZ- + zZX + xX-), 
viz. calling this (*]£X, Z) 4 , the coefficients (multiplying by 6) are 
(6w 2 , — 3y w, y 2 + 4xw, 6zw, 24w 2 ); 
and then writing 
L = y 2 + 4 xio, 
M = — 2 yz — 4 w 2 , 
we have 
N = — 4^ 2 + 16xw, 
i/ = L- — 12 w 2 M, 
— J=L 3 — 18 w 2 LM — 54 iv i N, 
and the equation is, as in former cases, 
X 2 (LN+M 2 ) - lHw-LMN - KhwXR - 21w i N 2 = 0 ; 
but LX + il/ 2 and therefore the whole equation divides by iu, and we thus obtain 
16L 2 (— xz 1 + y 2 x + w (yz + 4+) + iu 3 ) — 18 wLMX — 16wM 3 — 21w 3 X- = 0 ; 
or, completely developed, this is 
iu 7 .64 
+ w 5 .32 ( 3yz — 4x 2 ) 
+ w*. 16a; ( by 2 + 9z 2 ) 
+ w 3 . ( \f + 30y 2 z 2 + 160yzx 2 — 27z 4 + 64a- 4 ) 
+ w 2 . 4x (11 y 3 z + 12y 2 x 2 — 9yz 3 — 4+a; 2 ) 
+ iv . y' 2 ( y 3 z + 12y 2 x 2 — yz 3 — 8z 2 x 2 ) 
+ xfx ( y 2 - z 2 ) = 0.