506 
ON THE RECIPROCATION OF A QUARTIC DEVELOPABLE. 
[372 
This being so, the equation (1) belongs to a quartic torse, the reciprocal whereof 
is the skew cubic determined by the equations (4): and we have to show a posteriori 
that this is so. 
First if the torse is given, then the reciprocal figure is the envelope of the 
plane ax+ by + cz + diu = 0, (3), where (x, y, z, w) are the coordinates and (a, b, c, d) 
are regarded as variable parameters connected by the equation (1); we thence obtain 
the equations (2), where X is an arbitrary multiplier; and from the equations (1), (2), 
and (3), we have to eliminate a, b, c, d, X. The equation (3) is at once seen to be 
included in the equations (1) and (2); and the elimination would give only a single 
equation between the (x, y, z, w)—since however the equation (1) is that of a torse, 
we know that the elimination must give two equations, or more accurately a two-fold 
relation (represented, as in the present case it happens, by three equations) between 
the coordinates (x, y, z, w). 
Putting for shortness 
□ = a?d 2 — Qabcd + 4ac 3 + 4 b 3 d — Sb 2 c 2 : 
and substituting for Xx, Xy, Xz, Xw, their values from the equations (1), we have identically 
X 2 ( xz — \y 2 ) = — (bd — d) □, 
X 2 (iV z ~ xw) — — (ad — be) □, 
X 2 ( yw — \z 2 ) — — (ac — V 2 ) □; 
and hence (since □ = 0) we have between (x, y, z, w) the equations (4), showing that 
the reciprocal figure is a skew cubic. 
Secondly, let the skew cubic be given; then the reciprocal figure is the envelope 
of the plane ax -f by + cz + dw — 0, (3), where now (a, b, c, d) are the coordinates and 
(x, y, z, w) are regarded as variable parameters connected by the equations (4): we 
thence obtain the equations (5) containing the arbitrary multipliers p, q, r: and from 
the equations (3), (4), (5) we have to eliminate x, y, z, w, p, q, r. The equation (3) 
is at once seen to be included in the equations (4) and (5): since however the 
equations (4) are those of a curve, we know that the elimination must give a single 
equation between (a, b, c, d). 
The equations (4) are satisfied if we write therein 
x : y : z : w = £ : 6 : 6 2 : % 6 3 , 
and substituting these values of x, y, z, w in the equations (5), these equations give 
(a, b, c, d) in terms of 6, p, q, r, and we thence find identically 
ac — b 2 = — 6 2 (p — 2 qd + rd' 2 ) 2 , 
be — ad = — 26 (p — 2q6 + rd ) 2 , 
bd — c 2 = — (p — 2 qd + rd' 2 ) 2 , 
that is 
ac — b 2 : ad —be : bd—c 2 =d' 2 : — 2d : 1,