535] 
491 
535. 
NOTE ON THE PROBLEM OF ENVELOPES. 
[From the Messenger of Mathematics, vol. i. (1872), pp. 3, 4.] 
There is a mode of looking at the problem of Envelopes, which, so far as I am 
aware, has not been explicitly noticed. Let TJ = {x, y, z) m be a function of the 
coordinates {x, y, z), © = O' = (x, y, z) a (x, y, z') a ' a function of the two sets of coordi 
nates (x, y, z) and (x, y', z); it being understood that when we write © we regard 
(x, y, z) as the current coordinates, when ©' we regard {x, y, z) as the current 
coordinates. Suppose that we have TJ = 0; the curve ©' = 0 is then a curve the 
equation whereof contains as parameters the coordinates (x, y, z) of a point P on the 
curve TJ — 0; and we may seek for the envelope of the curve ©' = 0 as P describes 
the curve U = 0; the required envelope is of course obtained as an equation in (x\ y', z') 
given by the elimination of x, y, z, \ from the equations (equivalent to four equations 
only) 
U = 0, ©' = 0, 
d x ®' + \d x TJ - 0, 
dy © -f- \dy TJ — 0, 
<4©' + \d z TJ = 0. 
But, observe that the required envelope is the locus of the points of intersection 
of the curve ©' = 0 belonging to a particular point (x, y, z) of the curve TJ — 0, by 
the curve ©' = 0 which belongs to a consecutive point of TJ. The curve © = 0, con 
sidering therein (x', y, z') as the coordinates of a given point of the plane, determines 
by its intersection with [7 = 0 those points (x, y, z) on the curve [7 = 0, to each of 
which belongs a curve ©' = 0 passing through the point in question (x, y, z ). Hence, 
if the curve © = 0 touch the curve [7=0, the point of contact, coordinates (x, y, z), 
is a point such that to it and to the consecutive point there belong curves, each of 
them passing through the given point (x', y, z). Hence expressing that the curves 
62—2