ON A SPECIAL SEXTIC DEVELOPABLE.
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belonging to the point where the tangent line meets the cuspidal curve considered as
three coincident points, and to the points where it meets the nodal curve, are given
by the equation
(2b + at) 3 {b V2 — «Tj} (b Y2 — ar 2 } = 0,
that is
(2b + at) 3 (2b 2 — 2abt — a 2 t 2 ) = 0,
or say
(at + 2b) 3 (a 2 t 2 + 2abt — 2b 2 ) = 0.
I proceed to find the intersections of the tangent with the Prohessian: for this
purpose putting for a moment in the last-mentioned equation x for at and y for b, this is
(x + 2y) 3 (x 2 + 2xy — 2y 2 ) = 0,
or, if in the place of (x + y) we write x, this is
(x + y) 3 (x 2 — 3 y 2 ) = 0,
and the Hessian of this is easily found to be
(x + yY (Sx 2 + 8xy + 4y 2 );
whence, replacing x by (x + y), the Hessian of
(x + 2y) 3 ( x 2 + 2xy — 2y 2 ),
is
(x + 2y) 4 (Sx 2 + 14xy + 18y 2 ).
We have thus
that is
or
and therefore
b
a
or putting
Sx 2 + 14 xy + 18y 2 = 0 ;
Sx + {7 + V— 5} y = 0,
3at *f- [7 + V— 5j b = 0 j
-3 , — 3 {7 + V^~5} 4 _ 7 + V=T^
7+V-5 54 18
Wi
n 2
7 + V- 5
18 ’
7 -V- 5
18 ’
so that % + fi 2 = — b n i n 2 = b an d n 1} n 2 are the roots of the equation 18n 2 + 14?i -f 3 = 0,
then we have - = n 1 t or n 2 t, say -=nit, or assuming a = 1, then b = n 1 t.
Cb Cl
C. V.
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