ON A SPECIAL SEXTIC DEVELOPABLE. 
513 
373] 
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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