488 
ON THE DEVELOPABLE SURFACES WHICH ARISE FROM 
[84 
If, for shortness, we suppose 
F = YZ' - Y'Z, L =XW'-X'W, 
G =ZX' -Z'X, M= YW'-Y'W, 
H = XY' — X'Y, N = ZW — Z'W, 
(values which give rise to the identical equation 
LF + MG + NH = 0), 
then, A, p, v, p denoting any indeterminate quantities, the two linear equations in 
?/, £, co are identically satisfied by assuming 
£ = . N pi — Mv + Fp , 
p — — NX . -4- Lv (rp y 
f = Mx-Ip, . + Hp, 
co = — FX — Gp, — Hv 
and, substituting these values in the equations T = 0, T' = 0, we have two equations: 
A X 2 + B p? + G v 2 + 2jF p,v + 2 G vX + 2jR Xp, + 2L Xp + 2ilf pp + 2i\T vp = 0, 
A'X 2 + B'pr + C'v 2 + 2 F' pv + 2 G'vX + 2 H'Xp + 2L'Xp + 2 M'pp + 2 N' vp = 0, 
which are of course such as to permit the four quantities X, p, v, p to be simul 
taneously eliminated. The coefficients of these equations are obviously of the fourth 
order in x, y, z, w. 
Suppose for a moment that these coefficients (instead of being such as to permit 
this simultaneous elimination of X, p, v, p) denoted any arbitrary quantities, and suppose 
that the indeterminates X, p, v, p were besides connected by two linear equations, 
clX bp c v dp — 0, 
Q/X-\-bp-\-cv~\-dp = 0‘ } 
then, putting 
be' — b'c = f, ad' — a'd = l , 
ca — c'a — g, bd' — b'd = m, 
ab' — a'b = h, cd' — c'd = n , 
(values which give rise to the identical equation If + mg + nh = 0), and effecting the 
elimination of X. p, v, p between the four equations, we should obtain a final equation 
□ = 0, in which □ is a homogeneous function of the second order in each of the 
systems of coefficients A, B, &c., and A', B', &c., and a homogeneous function of the 
fourth order (indeterminate to a certain extent in its form on account of the identical