492 
ON THE DEVELOPABLE SURFACES WHICH ARISE FROM 
[84 
Proceeding next to the problem of finding the envelope of two surfaces of the second 
order, this is most readily effected by the following method communicated to me by 
Mr Salmon. Detaining the preceding notation, the equation U + kU'= 0 belongs to a 
surface of the second order passing through the Intersect of the two surfaces 
U — 0, U' — 0. The polar reciprocal of this surface U + JcU' = 0 is therefore a surface 
inscribed in the envelope of the reciprocals of the two surfaces U = 0, U' = 0, and 
consequently this envelope is the envelope (in the ordinary sense of the word) of the 
reciprocal of the surface U-\-kJJ' — 0, k being considered as a variable parameter. It 
is easily seen that the reciprocal of the surface U+kU' = 0 is given by an equation 
of the form 
A + 3B k + 3№ + D& 3 = 0, 
in which A, B, C, D are homogeneous functions of the second order in the coordi 
nates x, y, z, w. Differentiating with respect to k, and performing the elimination, we 
have for the equation of the envelope in question, 
(AD - BC) 2 - 4 (AC - B 2 ) (BD - C 2 ) = 0 ; 
or the envelope is, in general or (what is the same thing) for a system of the class 
(A), a developable of the eighth order. For a system of the class (B) the equation 
contains as a factor, the square of the linear function which equated to zero is the 
equation of the plane of contact; or the envelope is in this case a Developable of 
the sixth order. And in the case of a system of the class (G) the equation contains 
as a factor the cube of this linear function; or the envelope is a developable of the 
fifth order only. 
A. We may take for the two surfaces the reciprocals (with respect to x 2 + y 2 4- z 2 + w 2 = 0) 
of the equations made use of in determining the Intersect-Developable. The equations of 
these reciprocals are 
bcdx 2 4- c day 2 + dab z 2 + abcw 2 — 0, 
b'c'd'x 2 + c'd'a'y 2 + d'a'b'z 2 + a'b'c'w 2 = 0 ; 
and it is clear from the form of them (as compared with the equations of the surfaces 
of which they are the reciprocals) that x - 0, y = 0, z = 0, w = 0, are still the equations 
of the conjugate planes. We have, introducing the numerical factor 3 to avoid fractions, 
3 {(b + kb') (c + kc') (d + kd') x 2 + (c + kd') (d + kd') (a + ka') y 2 
+ (d + kd') (a + ka') (b + kb') z 2 + (a + ka') (b + Icb') (c + kc') w 2 ] 
= A + 3B& + 3C& 2 + D/c 3 , which determine the values of A, B, C, D. 
A = 3 (bcdx 2 + cday 2 + dabz 2 + abcw 2 ) 
B = (b'cd + bc'd + bed') x 2 + 
C = (bc'd' + b'cd' + b'c'd) x 2 + 
D = 3 (b'c'd'x 2 + c'd'a'y 2 + d'a'b'z 2 + a'b'c'w 2 ) 
We have in fact