514 
ON A SPECIAL SEXTIC DEVELOPABLE. 
[373 
But the equations of the tangent line being ut supra 
at 3 + 3 bt 2 + d = 0, 
Sat 4 + 8 bt 3 — e = 0, 
we have thus 
a = 1, 
b = n 2 t, 
d = (— 1 — 3ftj) ¿ 3 , 
e = ( 3 + 8nj) t 4 , 
b — n. 2 t, 
d — (— 1 — 3n 2 ) £ 3 , 
e = ( 3 + 8% 2 ) i 4 , 
as the coordinates of the required points, viz. the tangent line meets the Prohessian, 
in the point on the cuspidal edge considered as 6 points, in two points on the nodal 
curve and in the last-mentioned 2 points ; 64-2 + 2 = 10 the order of the Prohessian. 
The foregoing equations give 
ae — 6bd = 18w 1 2 + 14/q + 3 = 0, 
(in virtue of lSwd + 14n 2 + 3 = 0), 
so that the two points in question are the intersections of the tangent line with the 
surface ae — 6bd = 0. 
If we consider the intersection of this surface with the torse 
(ae — 4bd) 3 — 27 (— ad 2 — b 2 ef = 0, 
the equation ae — 6bd = 0, gives 
(ae — 4bd) 3 = (2bd) 3 = 8b 2 d 2 bd = £ aelrd?, 
and thence 
4 alrdre — 81 (ad 2 + b 2 ef = 0 ; 
that is 
81 a 2 d* — 158 a¥d 2 e + 815 4 e' 2 = 0, 
an equation which should agree with 
nrp 
We =&( 144^-23). 
In fact writing 
* = ^(144*1,-23), 
the equation 18/q 2 + 14^+3 = 0 is (18?q + 7) 2 + 5 = 0 ; that is, (144 n x + 56) 2 + 320 = 0, 
but 144 rq + 56 = 81a? +79, or the equation becomes (81a? + 79) 2 + 320 = 0, that is 
SI 2 « 2 + 81.158« + 81 2 = 0, or 81a? + 158 x + 81 = 0, 
which is right.