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A MEMOIR ON CUBIC SURFACES. 
417 
115. The cuspidal ninthic curve is a unicursal curve, the equations of which can 
be very readily obtained by considering it as the reciprocal of the spinode torse; we 
in fact have 
x : y : z : w = ZW+2XY : 2YZ + X 3 : WX + Y*-3Z* : ZX, 
or substituting for X, Y, Z, W their values (= 160 2 , 40 + 160 5 , 160 3 , -5— 80 4 —160 8 ) 
and omitting a common factor 166'-, we find for the cuspidal curve 
x : y : z : w = 30 + 240 3 -160 9 : 240 2 + 320 6 : -4-480 4 : 160 3 
(values which verify the equation Xx + Yy + Zz + Ww = 0); the spinode curve being 
thus of the order = 9 as mentioned. 
For 0 = oo we have the singular point (y = 0, z = 0, w = 0) (reciprocal of torsal 
biplane), and in the vicinity thereof x : y : z : w = 1 : — 20~ 3 : 30 -5 : — 6~ 6 , therefore 
For 0 = 0 we have the singular point a? = 0, y = 0, w — 0 (reciprocal of the other 
biplane), and in the vicinity thereof x : y : z : w = — f0 : — 60 2 : 1 : —40 s , therefore 
116. The section of the surface by the plane z = 0 is an interesting curve. 
Writing z — 0 in the equation of the surface, I find that the resulting equation may 
be written 
(64w 3 , 144xw-, w 3 +76x~w + xy-\w 2 + 27a? 2 , y 2 — 32ccw) 2 = 0, 
where observe that 
64w 3 (w 3 + 76x 3 w + xy-) — (72xw~)~ = 64w 3 \w {w- + 27a? 2 ) + a? (y 2 — 32xw)]; 
so that the curve has the four cusps w 2 + 27a? 2 = 0, y 2 — 32a?w = 0; the plane z = 0 
intersects the cuspidal ninthic curve in the point (y = 0, z — 0, w=0) counting 5 times, 
and in the last-mentioned four points: in fact, writing in the equations of the ninthic 
curve z = 0, that is 1 + 120 4 = 0, we find a?, y, w = §0, % 4 0 2 , 160 3 , and thence 
w~ + 27x? = %4.0 2 (1 +120 4 ) = 0, y 2 - 32xw = 0. 
The curve has also nodes at the points (y = 0, a? + w = 0 ; y = 0, a? — w = 0), viz. 
these are the intersections of the plane z = 0 with the nodal lines (y — z = 0, a? + w = 0) 
and (y + 2' = 0, a? — w = 0), and it has at the point (y = 0, w = 0) (intersection of its 
plane with the cusp-nodal line y = 0, w = 0, and quintic intersection with the cuspidal 
ninthic) a singular point = 2 cusps + 7 nodes; hence the curve has cusps = (4 + 2 =) 6 ; 
nodes (2 + 7 =) 9 ; or 2 nodes + 3 cusps = 36 ; class = 6, as it should be. 
C. VI. 
53