Mr. Airy on the Forms of the Teeth of Wheels. 285 
and in the case of trundles if it be required to take account of 
the diameter of the pins, this will be done by taking a curve, 
whose normal distance from the curve found by considering them 
as points, shall at all parts be equal to the radius of the pin. 
Or the form of the teeth may be found by the general theorem. 
For crown wheels, as the contrate wheel of a watch, the 
teeth without sensible error may have the same form as for rack- 
work. The theory may be extended to bevelled wheels, without 
any difficulty. 
There is one case which ought to be mentioned particularly. 
It may be desired that the teeth of one wheel have plane surfaces 
passing through the axis of the wheel. Since a straight line is 
the hypocycloid, in which the radius of the generating circle is 
half that of the fixed circle, the teeth of the other wheel must 
be epicycloids, the radius of the generating circle being half that 
of the first wheel. The action here takes place entirely after the 
line of centers, and the direction of the action is nearly per- 
pendicular to that line. I imagine this to be a good construction 
for pinions with a small number of teeth driven by a large wheel. 
If each tooth consist of a line within the principal circle, and 
an epicycloid without it, the radius of the generating circle of 
each epicycloid, being half that of the other principal circle, 
a very good form will be produced. The action takes place 
before as well as after the line of centers, and is always nearly 
perpendicular to that line. The figure usually given to the teeth 
of watch-wheels approaches very nearly to this. 
I have confined my attention entirely to uniformity of action, 
and uniformity of motion, as I conceive them to be of far greater 
consequence than the diminution of friction. The friction can 
never be made = 0, except the point of contact be always in the 
line of centers ; a condition which may be satisfied by an infinite 
number of curves, and amongst others by two logarithmic spirals. 
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