104 
ON A CERTAIN SEXTIC TORSE. 
[436 
The foregoing results lead to the conclusion that for w= 0, we have 
A — (g-h?x + h 2 f 2 y + pg 2 zf \{g?x + b-y + c 2 z) 3 — 27a 2 b 2 c 2 xyz] ; 
but this will appear more distinctly as follows. 
7. First, as to the factor {a?x + bhy + c-zf — 27a 2 b 2 c 1 xyz : writing in the equation of 
the osculating plane w = 0, the equation becomes 
which equation is therefore that of the trace of the osculating plane on the plane 
w = 0; the envelope of the trace in question is a part of the section of the torse by 
the plane iu = 0. To find the equation of this envelope we must eliminate 6 from the 
foregoing, and its derived equation 
the two equations give 
x : y : z=a{6 + a) 3 : b (9 + (3) 3 : c(0+ry) 3 , 
(ia?xf + (Jf-yŸ + {c 2 zf = a (6 + a) + b (6 + /3) + c (0 + 7) = 0, 
and thence 
that is, we have 
(a 2 xf + (b-yf + (cV) :i = 0, 
or, what is the same thing, 
(a?x + b 2 y + c 2 z) s — 27 a 2 b 2 c 2 xyz = 0 
for a part of the section in question. 
8. I have said that the foregoing cubic is a part of the section ; the equations 
x : y : z : w = ahg (6 + a) 4 : bhf(d + fiy : cfg(0 + yY : abc (6 + 8) 4 , 
which for w = 0 give 6 = — 8, and thence x : y : z = ap : bg 3 : ch 3 , show that the last 
mentioned point is a four-pointic intersection of the curve with the plane w = 0. 
But the curve, having four consecutive points, will have three consecutive tangents in 
the plane w = 0 ; that is, the tangent at the point in question will present itself as 
a threefold factor in the equation of the torse. Writing in the equations of the tangent 
= 0, 6 = — 8, we find for the equation of the tangent in question 
or, what is the same thing, 
g 2 h 2 x + h 2 f 2 y +pg-z = 0. 
Hence the section by the plane w = 0 is made up of this line taken three times, 
and of the last mentioned cubic curve.