# Page:The Kinematics of Machinery.djvu/185

163
CENTROIDS AS PROFILES.

of twelve teeth the flanks become radii. This occurs if O C′ = the pitch x 6/π sin 75°. A closer examination shows this construction to be identical with that given in Fig. 116, when equal constants are used. RC′and RC″ are the diameters of our auxiliary centroids, which are here dotted so that the constructions may

Fig. 118.

be compared; the points C′ and C″ correspond to of a′ and b′, P and Q are found as before. Willis[1] himself recognised and mentioned this coincidence.[2]

§ 37
Seventh Method.—The Centroids themselves as Profiles of Elements.

If we suppose the auxiliary centroids of the method of § 32 to be closed figures, and to be made smaller and smaller until they

1. Principles of Mechanism, 1st Ed. p. 107, 2nd Ed. p. 142.
2. The approximation proposed by Professor Unwin, and described by him in Engineering, May 29, 1874, is perhaps more exact than either of these. He finds two points in each roulette (epi- or hypo-cycloid), one upon the pitch circle, and one
M 2