504 
ON THE DEVELOPABLE DERIVED FROM 
[86 
or the equations of the edge of regression are given by the system (equivalent of 
course to two equations), 
|| a, /3, 7, A, B !| = 0. 
I ¡3, 7 , 8, B, G | 
The simplest mode of verifying a 'posteriori that the edge of regression is only of 
the ninth order, appears to be to consider this curve as the common intersection of 
the three surfaces of the seventh order: 
A s a - 3A 2 Bb + 3AB 2 c - B s d = 0, 
A s b - 2>A 2 Bc + 3A B 2 d - B 3 e = 0, 
^l 3 c - 3A 2 Bd + 3AB 2 e - B : f = 0, 
(which are at once obtained by combining the equation Bt + A = 0 with the cubic 
equations in t). It is obvious from a preceding equation that if the equations first 
given are multiplied by fA, — 3 eA + fB, 2d A — eB, and added, an identical result is 
obtained. This shows that the curve of the forty-ninth order, the intersection of the 
first two surfaces, is made up of the curve in question, the curve of the fourth order 
.4=0, B = 0 (which reckons for thirty-six, as being a triple line on each surface), and 
the curve which is common to the two surfaces of the seventh order and the surface 
2d A — eB = 0. The equations of this last curve may be written, 
e (af— 3 be + 2 cd) — 4 d (bf— 4 ce + 3 d 2 ) — 0, 
e s a — 6e 2 c?6 + 12 ed 2 c — 8 d* = 0, 
e 3 b — Qe-dc + 4 ed z — 0 ; 
or, observing that these equations are 
f (ae — 4<bd) — 3 (be 2 — 6ced + 4c? 3 ) = 0, 
e 2 (ae — 46c?) — 2c? (be 2 — 6ced + 4c? 3 ) = 0, 
e (be 2 — 6ced + 4c? 3 ) = 0 ; 
the last-mentioned curve is the intersection of 
ae — 46c? = 0, 
be 2 — 6ced + 4c? 3 = 0, 
where the second surface contains the double line e = 0, d = 0, which is also a single 
line upon the first surface. Omitting this extraneous line, the intersection is of the 
fourth order; and we may remark that, in passing, it is determined (exclusively of the 
double line) as the intersection of the three surfaces 
ae — 46c? = 0, 
be 2 — Seed + 4c? 3 = 0, 
a 2 d — 6abc + 46 3 = 0,