490 
ON THE DEVELOPABLE SURFACES WHICH ARISE FROM 
[84 
where x — 0, y = 0, z — 0, and w — 0, are the equations of the four conjugate planes. 
There is no particular difficulty in performing the operations indicated by the general 
process given above; and if we write, in order to abbreviate, 
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 satisfy the identical equation If + mg + nh = 0), the result after all 
reductions is 
pfiyigi _|_ _|_ nddxf 4 + l i f-xdiv i + ndfyhv 4 4- nddzdud 
+ 2mng 2 h 2 oc i y 2 z 2 + 2nlh 2 f 2 y 4 z 2 x 2 + 2lmf 2 g 2 z 4 x 2 y 2 
— 2m 2 n 2 ghw 4 y 2 z 2 — 2n 2 l 2 hfw 4 z 2 x 2 — 2l 2 rnfgw 4 x 2 y 2 
+ 2fmg 2 l 2 x i z 2 w 2 + 2gnh-m 2 y i xhv' 2 + 2hlf 2 n 2 z i y 2 w 2 
— 2fnli 1 Px i yhv' 2 — 2glf 2 rr?y i z 2 w 2 — 2 hmg 2 n 2 z i xhu 2 
+ 2 (mg — nh) (nh — If) (If — mg) x 2 y 2 z 2 w 2 = 0, 
which is therefore the equation of the Intersect-Developable for this case. The discussion 
of the geometrical properties of the surface will be very much facilitated by presenting 
the equation under the following form, which is evidently one of a system of six 
different forms, 
{;m (gx 1 + mv 2 ) (hy 2 — gz 2 + lw' 2 ) - l (—fy 2 + nw 2 ) (— hx 2 4 fz 2 4- mw 2 )) 2 
— 4fglmx 2 y 2 (hy 2 — gz 2 4- lw 2 ) (— hx 2 4- fz 2 4- mw 2 ) = 0. 
B. Two surfaces forming a system belonging to this class may be represented by 
equations such as 
a x 2 + b y 2 + c z 2 + 2n zw = 0, 
a'x 2 4- b'y 2 + c!z 2 4- 2n'zw = 0, 
in which ¿c = 0, y= 0 are the equations of the single conjugate planes, z=0 that of 
the double conjugate plane or plane of contact, w = 0 that of an indeterminate plane 
through the two single conjugate points. If we write 
be' — b'c = f, an' — an =p, 
ca' — c'a = g, bn' —b'n —q, 
ab' — a'b = h, cn' — c'n = r, 
(values which satisfy the identical equation pf+qg + rh = 0), the result after all reduc 
tions is 
i A h 2 z 6 + 2pr 2 h (rli — qg) z 4 x 2 — 2qr 2 h (pf— rh) z i y 2 
4_ 4,p 2 qr 2 hz 3 x 2 w — 4<pq 2 r 2 hz 3 y 2 w 
+p : (rh — qg) 2 z 2 x A 4- q 2 (pf — rh) 2 z 2 y A 4- 2pq (4r 2 li 2 — fgpq) z 2 x 2 y 2 
+ 4p s q (rh — qg) zx 4 w 4- 4pq s zy 4 w — 4p 2 q 2 (qg —pf) zx 2 y 2 w 
+ 4p 4 q 2 x A w 2 4- 4p 2 q 4 y 4 w 2 4- 8p 3 q 3 x 2 y 2 w 2 + kp 2 qrh 2 ody 2 4- 4pq 2 rh 2 x 2 f = 0,