On Cog or Toothed Wheels. 151 
at that point. Let A and B be two circles, 
tangent to each other (Fig. 3) in e. AC isthe 
line joining the centres, and DF the common 
tangent of the circles at e; which is at right 
angles with AC ; and so are the circumferen- 
ces of the two circles at the point e.. For 
the circles and tangent coincide for the mo- 
ment. Hence then I conclude, Ist, that a 
motion (evanescently small) of the point com- 
mon to the three lines, can take place with- 
out quitting the tangent DF: and 2d, that if 
there is an infinite number of teeth in these 
circles, those which are found in the line of 
the centres, will geer together in preference 
to those which are out of it, since the latter 
have the common tangent, and an interval of 
space between them. 
The truth of this proposition (or an “hia 
finite approximation to truth,) may be dedu- 
ced from the supposition that the two circles 
do actually penetrate each other. To this end 
let AB, ab, in fig. 5, be two equal cir cles, 
placed parallel te each other in two contigu- 
ous planes, so as for one to hide the other, 
in the indefinitely small curvilinear space 
dfeg. I say that if the arc dg is indefi- 
nitely small, the rotation of the two circles 
will occasion no more friction between the 
touching surfaces, ge f and fd g, than there 
