XVIII. On the Forms of the Teeth of Wheels. 
By GEORGE BIDDELL AIRY, B.A. 
FELLOW OF TRINITY COLLEGE, AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 
AND CORRESPONDING MEMBER OF THE NORTHERN INSTITUTE. 
[Read May 2, 1825.] 
Tue investigation of the forms proper for the teeth of 
wheels is a useful and interesting inquiry. The mechanical prin- 
ciples are very simple, and the geometrical propositions on which 
it is immediately made to depend, admit of being put in an 
elegant form. But all the theories which have yet been given, 
are, I believe, very imperfect. Euler in the New Petersburgh 
Commentaries for 1760 has treated the subject with great gene- 
rality; but the analytical method which he has used is very 
unfavourable for the discovery of the most obvious. properties 
of the curves. In all the other theories that I have seen, no 
-forms are mentioned but the involute of a circle, and the epi- 
cycloid and hypocycloid. In this paper I propose to consider 
generally the figures which must be given to the teeth of wheels 
to insure uniformity of action. The curves above alluded to, 
though probably the most convenient of all, I shall shew are 
particular cases of a very general construction: and the demon- 
stration which has usually been given for them, I shall apply 
to every other case. . 
That the mechanical effect which one wheel produces upon 
another, may in all positions be the same, it is necessary that 
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