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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. *R*′ *C*′and *R*″ *C*″ are the diameters of our auxiliary centroids, which are here dotted so that the constructions may

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

- ↑
*Principles of Mechanism*, 1st Ed. p. 107, 2nd Ed. p. 142. - ↑ 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

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