594
�NOTES.
�19 (P. 145.) Open square figures, like ABGD, give square point paths If
the centroid of the curved disc be a duangle (compare PI. XII. Fig. 1). This
occurs here, however, if QT be normal to PS. For we have, by similar
T)T>
triangles, Z_lPQ=/^m 1 SP, and hence m^ P= ^- cos 1PQ=PS sin 1PQ, or (as
PR = PS), tan 1PQ=. The form of the disc must therefore be so chosen that the tangent of the angle m l SP=0'5, or that m L P=O5 m l S. [I regret that the Figures on PI. XII. have been transposed by mistake. Fig. 1 is referred to in the text as Fig. 2, and vice versa.]
20 (P. 154). Willis (Elements of Mechanism, 2nd Edition, p. 90) gives Camus' theorem as follows :
" If the pinion is to turn the wheel with a uniform force, the curve of its leaf, and that of the tooth of the wheel must be generated in the manner of epicycloids by one and the same describing curve, which must be rolled within the circle of the pinion to describe the inner form of the leaf, and on the outside of the circle of the wheel to describe the outer form of the tooth," etc. [Willis gives the quotation from Camus also in the original.]
[It is very interesting to compare the " solutions " given in Willis's third chapter with this portion of Reuleaux's work. Willis gives the method of 32 as the " general solution," his other solutions are all for the special cases of circular centroids. With this limitation the first solution corresponds with 34, the second and third are special cases of 32, and the fourth corresponds with 33. In a subsequent section he uses the method of 35 for pin teeth.]
21 (P. 154.) Ann. Ph. 1706, p. 379. De la Hire enunciates the proposition as follows : " It is always possible to find a curve which, by revolving upon a given base-curve, shall generate by some describing point, in the manner of
a trochoid, a second given curve ; provided that the normals from all points of the second curve meet the first." He gives as an illustration the production of a straight line by rolling a curve upon a second straight line cutting the first. The describ- ing curve, as can be easily seen, is a loga- rithmic spiral. If the lines are parallel the spiral becomes a circle.
22 (P. 171.) Suppose the two pieces a and &, Fig. 446, to be pressed together FJG 440 by some force, P, normal to their surfaces,
and that these are covered with regularly
formed teeth whose sides are inclined at a uniform angle, 0, in the way shown in the figure ; the resultant, Q, of the pressures upon the sides of the teeth opposed to any intended motion resolves itself into the resistance, P to the load and the resistance, F, to the force tending to produce sliding. F therefore is = P tan <. We could in this way determine, by experiments on the " friction of rest," the mean angle of the roughnesses upon the two eurfaces in contact.
I may be permitted to take this opportunity of pointing out the inexactness
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