wheels;* any pair of them, equally pitched, will gear truly together. Willis ) appears to have been the first to point out both this problem and its solution. Looking at it from the general point of view which we have here reached, it is evident that the solution of the problem is the use of similar auxiliary centroids for those shown on opposite sides of the primary centroids in Fig. 108.
The delineation of wheel-teeth very early led geometricians to the use of roulettes as profiles of elements. Camus laid down its fundamental principles very clearly in 1733 in a little-known treatise, as Willis has shown. 20
Camus' predecessor, De la Hire, had apparently used this method before him, and he himself refers to the still earlier Desargues J (1593 1662) as having used epicycloidal teeth, and thus preceded by many years Romer (1664 1710), so often mentioned as their inventor.
In the distinctness and completeness of its results, this method of forming element profiles by roulettes greatly excels the method first described, which indeed, in a certain sense, it includes. For we may conceive of the assumed profile of the first method as having been itself found by means of an auxiliary centroid. The second profile then becomes a roulette drawn by the same curve, which thus may be considered as actually the describing curve of the profiles, although it has not itself been drawn.
De la Hire also enunciated the general proposition as to the describing of roulettes which we are here applying, and which is of so great importance in Kinematics; and it is to the same geome- trician that we owe their name. 21 The methods and propositions relating to them have hitherto hardly received their due develop- ment. The delineation of the auxiliary centroid in the method of 31 is interesting but not necessary, the method itself is prac- tically useful chiefly where a single result is all that is required.
bodies to which they belong will have different centroids from those originally assumed. Thus in the special case mentioned above, if the wheel teeth be not of the right shape, the wheels will not have a constant, but a varying angular velocity ratio. Their centroids will therefore no longer be circles, but irregular figures more or less nearly resembling them.
- I cannot find that any name has hitherto been used for them in this country. In
Germany they are called Satz-rader.
t Trans, of Inst of Civil Engineers, 1837, vol. ii. p. 91.
J Chasles, Qescliiclite der Geometric, p. 83. (Sohncke.)
