284 Mr. Airy on the Forms of the Teeth of Wheels. 
considered as nearly representing the motion of the circumference.) 
Also the pressure occasioned by a given force in given cireum- 
1 : $3 415% 
stances * — pap: and the mechanical effect of friction is propor- 
tional to the pressure by which it is caused multiplied by the velocity 
of the rubbing surfaces; and therefore > aay nearly. The 
numerator is proportional to the distance from the line of centers; 
and therefore will be the same for all teeth, when that distance is 
the same. But the denominator is largest when the face of the 
tooth is parallel to the radius of the circle. I imagine then that 
it is advisable to make the teeth work a little before as well as 
a little after the line of centers. And I should think that a tooth 
similar to that formed by the union of the epicycloid and hypo- 
eycloid, is preferable to any other form whatever. For the line 
of action is always very nearly perpendicular to the radius ; by which 
means not only is the friction made much less, but also the strain 
upon the axes is considerably diminished. 
If it be thought desirable to prevent back-lashing, this can 
be done by giving proper forms on the same principles to the 
faces of the teeth, which are not the working faces. But the 
chance of very greatly increasing the friction, makes the propriety 
of this consideration very doubtful. 
The whole of what has been stated with regard to circles, 
it is evident will apply equally to straight lines. Thus the teeth 
of rack-work may be formed of a combination of cycloids, in 
which case those of the wheel must consist of epicycloids, and 
hypocycloids; they may be straight, which will make those of 
the wheel the involutes of a circle, (both being generated by the 
revolution of a logarithmic spiral ;) they may be mere pins, in 
which case the teeth of the wheel will be involutes, or curves 
described in nearly the same manner as involutes. In this case, 
