510 ON THE RECIPROCATION OF A QUARTIC DEVELOPABLE. [372 
I remark that in general, if D=<j> (a, b, c, d) = 0 is the equation of a torse, then 
for finding the reciprocal curve, we have 
□ = 0, ax + by 4- cz + dw = 0, 
8 a O 4- \x — 0, 
+ \y — 0, 
8 C CH + \z — 0, 
+ \w = 0, 
and that from these equations we deduce not only 
a8x + b8y + c8z + d8w = 0, 
but also the equation 
(21, 33, 6, 3), & ©, £, 2, 93?, 9?$a, ft 7 , %, 8*, 8w) = 0, 
where 21,... are the inverse system derived from the second differential coefficients 
of □ : (a, ft 7, 8) are arbitrary coefficients, introduced only for symmetry, and there 
is no real loss of generality in reducing all but one of them to zero, and so reducing 
the equation for example to the form 
+ $8y + ®8z + %8iv = 0. 
The existence of the two linear equations between (8x, 8y, 8z, 8w) proves that (x, y, z, w) 
are connected by a two-fold relation, that is, that the reciprocal of the given torse 
is a curve. 
Cambridge, January 26, 1865.