ON THE DEVELOPABLE SURFACES WHICH ARISE FROM 
494 
[84 
which is therefore the equation of the envelope for this case. The equation may also be 
presented under the form 
w' 2 B + (-mnx 2 4- nly- + Imz 2 ) 2 (/ve 4 + g 2 y* + h-x A — 2ghy 2 z 2 - 2hfz 2 x 2 — 2fgx 2 y 2 ) — 0 ; 
and there are probably other forms proper to exhibit the different geometrical properties 
of the surface, but with which I am not yet acquainted. 
B. Here taking for the two surfaces the reciprocals of the equations made use 
of in determining the Intersect-Developable, the equations of these reciprocals are 
n- b x 1 + n 2 ay 1 — ah c w 2 + 2n abzw = 0, 
n' 2 b'x 2 + n' 2 a'y 2 — a'b'c'w 2 + 2 n'a'b'zw — 0, 
which are similar to the equations of the surfaces of which they are reciprocal, only 
z and w are interchanged, so that here x = 0, y = 0 are the single conjugate planes, 
z — 0 is an indeterminate plane passing through the single conjugate points, and w = 0 
is the equation of the double conjugate plane or plane of contact. 
The values of A, B, C, D are 
A = 3 (tfbx 2 + ri 2 ay 2 — abciv 2 + 2 nabzw), 
B = (2nn'b + n 2 b') x 2 + 
C = (2nn'b' + n' 2 b) x 2 + 
D = 3 (n' 2 b'x 2 + n' 2 a'y 2 — a'b'c'w 2 + 2n'a'b'zw). 
Hence, using f, g, h, p, q, r in the same sense as before, we have for 
2 (AC — B 2 ), 2 (BD — C 2 ), (AD — BC) expressions of the same form as in the last case 
(p, q, r being written for l, m, n), but in which 
A = f 2 w 4 -f 4 q 2 z 2 w 2 + 8 qry 2 w 2 — 4fqzw 3 , 
B = ghv 4 + 4p 2 z 2 w 2 + 8prx 2 w 2 + gpzvf, 
C = /¿ 2 w 4 , 
D = 2 q 2 x? + 2p 2 y 4 + 4 h 2 z 2 w 2 + 4pqx 2 y 2 — 8 qhx 2 zw + 8phy 2 zw + 2ghy 2 w 2 + 2fhx 2 w 
F =-p 2 y 2 w 2 , 
G = — q 2 x 2 w 2 , 
H= 0, 
L = 2pqifzw, 
M = 2pqx 2 ziv, 
N = — 2 h 2 zw z .