278 
ON CERTAIN DEVELOPABLE SURFACES. 
[344 
which give 
(ac — b 2 ) {3c (ae — c 2 ) — 4e (ac — b 2 )} = 0, U= 0, 
and these are equivalent to 
(ac — b 2 = 0, 27= 0) and {3c (ae — c 2 ) — 4e (ac — b 2 ) — 0, 27 = 0}, 
so that the entire intersection is made up of (ac -b 2 = 0, 27= 0) twice, and of 
{4e (ac — b 2 ) — 3c (ae — c 2 ) = 0, U = 0} once. 
25. The first part is at once seen to give 
the cuspidal curve (ac — b 2 = 0, ae — c 2 = 0) 4 times, order 16 
the line (a = 0, b = 0) 4 „ „ 4 
20 
The second part gives 
(ae — c 2 ) {4c (ac — b 2 ) + a (ae — c 2 )} = 0, 
{4e (ac — b 2 ) — 3c (ae — c 2 )} = 0, 
this consists of 1° the part ae — c 2 = 0, e(ac — b 2 ) = 0, viz. 
the cuspidal curve (ac — b 2 = 0, ae — c 2 = 0) once, order 4 
the line (c = 0, e = 0) 
twice, 
and 2° the part 
4c (ac — b 2 ) + a (ae — c 2 ) = 0, 
4e (ac — b 2 ) — 3c (ae — c 2 ) = 0, 
which contains 
the cuspidal curve (ac — b 2 = 0, ae — c 2 = 0) once, order 4, 
and by writing the two equations in the form 
c (ae + 3c 2 ) — 4b 2 e = 0, 
a (ae + 3c 2 ) — 4b 2 c = 0, 
it is clear that it contains also 
the nodal curve (ae + 3c 2 , 5 = 0) twice, order 4 
and the line (c = 0, e = 0) once, „ 1 
T 
whence the complete intersection of the developable and the Prohessian is made up 
as follows, viz. 
the cuspidal curve (ac — b 2 = 0, ae — c 2 = 0) 6 times, order 24 
the nodal curve (ae + 3c 2 = 0, b = 0) 2 times, „ 4 
the line (a = 0, b = 0) 4 times, „ 4 
the line (a = 0, e = 0) 3 times, „ 3 
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