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PROFESSOR CATLET ON CUBIC SURFACES. 
79. Combining with the foregoing the equation of the surface 
XZW+Y 2 (yZ+W)+(a, b, c, dJX, Y) 3 =0, 
it appears that these have along the line X=0, Y=0 the common tangent plane X=0, 
or, what is the same thing, that they meet in the line X=0, Y=0 (the axis) twice, and 
in a residual curve of the tenth order, which is the spinode curve ; the equations may be 
presented in the somewhat more simple form 
XZW+Y 2 (yZ + &W)+(a, b, c, dJX, Y) 3 =0, 
-4y&Y 2 ZW-4(yZ+W)(a, b, c, dJX, Y) 3 +12y&XY 2 («X+&Y) 
+ X 4 ( — 1 2«c + 95 2 ) — 3 cZ(4«X 3 Y + 6 &X 2 Y 2 + 4cXY 3 + dY*) = 0 , 
which, however, still contain the line X=0, Y=0 twice. The spinode curve, as just 
mentioned, is of the tenth order; that is, we have <7'=10. 
Each of the 6 mere lines is a double tangent to the spinode curve, but the trans- 
versal is only a single tangent : to show this, observe that the equations of the trans- 
versal are X=0, yZ+SW+<ZY=0 ; substituting in the equations of the curve the first 
equation, that of the cubic surface is of course satisfied identically ; for the second 
equation, writing X=0, this becomes Y 2 { — 4y£ZW— 4dY(yZ + rfVV) — 3d 2 Y 2 }=0 ; or 
writing herein dY= — (yZ+£W), it becomes Y 2 ( y Z — 0 W ) 2 = 0 . The value Y 2 =0 gives 
X=0, Y=0, yZ+c$W=0, viz. this is a point on the axis X=0, Y=0 not belonging to 
the spinode curve; the value (yZ — £W) 2 =0 gives a point of contact X=0, 
yZ+SW -j-dY=0, yZ— &W=0; and the transversal is thus a single tangent. Hence 
the number of contacts is 2.6+1, =13 ; that is, we have /3'= 13. 
Reciprocal Surface. 
80. The equation is found by equating to zero the discriminant of the binary quartic 
{£X 2 +yXY-(^ + yw)Y 2 }+4Zw{X(«, b, c, dJX, Y) 3 -ySY 4 }, 
or say this is (#)[X, Y) 4 , where the coefficients are 
6# 2 +24«zw, 
3 xy +18 bzw, 
y 1 — 2 (&z + y w)x + 1 2czw, 
— o(bz-\-yw)y + 6dzw, 
6(fe— yw) 2 . 
81. Forming the invariants, these are 
+[=A 2 + 24Uzw + 144^W 
— J= A 3 + 36AUm’+216V2 2 w 2 + 864^ 3 w 3 , 
