398] and sextic surface connected therewith. 97 
and we have 
X = 2(^ - S)(7 - 8)( 7 - a) (a - /3), 
/x = 2 (7 — 8) (a - 8) (a-/3) (/3- 7 ), 
v = 2 (a - 8) (Ji - 8) (/3 - 7) (7 - a ), 
whence X/xx» = 8X'/xV. 
Hence finally X, /x, y denoting as just mentioned, and therefore satisfying 
- + - + - = 0, the equation of the developable is 
X /x v 
Xpv [iv 2 — w (x 2 + y 2 + z 2 ) + 2xyz\ 2 + 108 {(X + /x + v) w 2 — Xa 2 — /xy 2 — vz 2 } 3 = 0 
(say this is X/jlvT 2 + 108$ 3 = 0), and this surface (which has obviously the cuspidal curve 
S = 0, T — 0) has also the nodal curve 
wxyz = 
x 1 (/xy 2 — i^ 2 ) _ y' 1 (vz 2 — Xx 2 ) _ z 2 (X« 2 — /xy 2 ) 
y — v v — X X — fM 
I will show cl 'posteriori that this is actually a nodal curve on the surface. Intro 
ducing an arbitrary parameter 0, the equations of the curve may be written lit supra 
0Xx = xw — yz, 
dfiy = yw — zx, 
Ovz =zw — xy, 
20 (X + /x + v) w — Sw 2 — x 2 — y 2 — z 2 , 
and we have thence, as before, 
2w Uw _ _ " _ Sfl -*. 
\ X y z) ° 
Hence 
2$ _ ^ tL>2 — x ‘ — y* — z? 2w (x 2 -f y 2 + z 2 ) + Qxyz 
(X + /jl + v) iv — Xx 2 — /xy 2 — vz 2 
_ 3 w s —w(x 2 +y 2 + z 2 ) 
(X + /x + v) w 2 * 
_ 3 [w 2 — w (x 2 + y 2 + z 2 ) + 2 xyz] 
(X + /x + v) w 2 — Xx 2 — /xy 2 — vz 2 ’ 
_ 3 T 
~ S • 
Hence writing 
0 (— 2Xx) — (- 2wx + 2yz) = 0, 
0 (— 2/xy) — (- 2 wy + 2 zx) =0, 
6 (— 2vz) - (- 2m -I- 2xy) = 0, 
0.2(X + /x+i/)w- (3w 2 -oc 2 -y 2 — z 2 ) = 0, 
C. VI. 
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