734] 
103 
734. 
ON THE KINEMATICS OF A PLANE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 
pp. 1—8.] 
It seems desirable to bring together under this title various questions which 
have been, or may be, proposed or discussed. We consider two planes in relative 
motion one upon the other, but, for convenience, they may be distinguished as a 
moving plane and a fixed plane, the first moving upon the second. Any point of 
the moving plane traces out on the fixed plane a curve, and any line of the moving 
plane envelopes on the fixed plane a curve; similarly, any point of the fixed plane 
traces out on the moving plane a curve, and any line of the fixed plane envelopes 
on the moving plane a curve. More generally, any curve of the moving plane envelopes 
on the fixed plane a curve, and any curve of the fixed plane envelopes on the 
moving plane a curve. There is, moreover, in the moving plane a curve which rolls 
upon a curve in the fixed plane, and these two curves (a single relative position 
being given) determine the motion. 
Fig. 1. 
The analytical theory presents no difficulty. Taking in the fixed plane the fixed 
axes Ox, Oy (fig. 1), and, fixed in the moveable plane so as to move with it, the 
axes OjXj, 0 1 y 1 ; then the position of the axes 0 1 x 1 y 1 may be determined, say by