Mr. Airy on the Forms of the Teeth of Wheels. 283 
a is >90°, r may be positive or negative, and must be greater 
than the radius in the same direction. 
If then, as is the case in general, a be < 90°, that part 
of the tooth which is without the circle, must be formed by the 
revolution of some curve upon the circle, and that which is within 
it by the revolution of some curve within the circle. This kind 
of tooth is represented in Fig. 4. But if a may be > 90’, 
the whole of the teeth may be formed by the revolution of a 
single curve; an instance of this is represented in (Fig. 8.) where 
the teeth GH and KL are formed by the motion of MN, carrying 
the describing point P. In the last case, if the curve be a circle 
equal to one of the circles, one tooth will be reduced to a point, the 
other will be an epicycloid or epitrochoid, according as the describing 
point is in the circumference of the circle, or in any other part. 
It will easily be seen, that where the acting surface of the 
driving tooth is above the cirele, the action takes place after 
passing the line joining the centers; when below the circle, it is 
before passing that line. Now practical men always think it 
proper, that the action should take place only after passing the 
line of centers. It is thought necessary that the direction of the 
friction should be such as to wipe off the dust, &c. from the teeth. 
For this purpose then, the curve which has been found for the 
lower part of the teeth, must be considered as a limit which 
that tooth must not reach. In the case in which the whole is 
fermed by the revolution of one curve, the whole action takes 
place after passing the line of centers. 
To find what the friction really amounts to, we have merely 
to observe, that in Fig.1. if D be brought to G in one tooth, 
and to F in the other, GF is the friction, and if BDC =a, FG : 
FD :: sn ADB : sin a; therefore frictional motion « ee 
sin ADB j 
BCD nearly, (the teeth being so small, that DF may be 
Vol. Ll. Part U. Oo 
