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It may be remarked that the wheel AC is called the driver, and вс the follower.

Ex. 542. If in the last Article a single wheel moving on a parallel axle with its centre in the line An were interposed between AC and BC, it would cause the follower to revolve in the same direction as the driver, and would not produce any change in the ratio of their angular motions, the radii AC and BC being unchanged.

107. Practical Objection to the above Solution. It is evident that the above solution fails if the surfaces of the wheels rub smooth, so that the motion becomes partly one of sliding and partly one of rolling contact; and also that it will fail if the centres A and B are slightly displaced, since then the contact ceases: one method, in common use, of obviating this objection is to pass a strong band of leather tightly over the wheels; this method is commonly used when the centres A and B are so considerable a distance apart that the wheels would be inconveniently large if in immediate contact; the most effectual means, and the only one with which we are here concerned, is to cut teeth on the circumferences of the wheels; when this is properly done the uniform revolution of the wheel A can be made to communicate a uniform revolution to the wheel в. The problem we are to solve is therefore twofold :

(1) To determine the form that must be given to the teeth of wheels, in order that any uniform motion of the driver round its axis shall communicate to the follower a uniform motion round its axis.

(2) As this cannot be done without causing the teeth of the one wheel to slide over those of the other, it is required to determine what amount of work is lost by the friction of the teeth when work is transmitted from one axle to the other.

The limits of the present work will not allow us to do more than give one solution of the former question, and an approximate solution of the second. Readers who desire further information on this very important subject will be able to obtain it by reference to Mr. Willis's Principles of Mechanism,' and to Mr. Moseley's 'Mechanical Principles of Engineering:'* the former work treats only of the question of form; the latter also contains a very full discussion of the question of force.

FIG. 147.

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108. Definition and Properties of the Epicycloid.If a circle carrying on its circumference a pencil-point be made to roll on the outside of the circumference of fixed circle, the point will

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called an epicycloid : the fixed circle is called the base; the moving circle is called the generating circle. Thus if qis a point

on the generating circle A D Q, and APC is the base or fixed circle, then if q were in contact with APC at P, the point Q will trace out the epicycloid P Q.

(a) It is evident that the length of the arc aq equals

that of the arc A P.

(b) It is evident that the point q is at the instant moving in a circle of which the centre is A, and radius AQ, so that the line aq is the normal to the epicycloid at the point q, and if DQ be joined that line is a tangent to the curve at Q.

(c) It is evident that the form and dimensions of the curve are independent of the particular point Q occupies

* A very clear elementary discussion of the forms of the teeth of wheels will be found in Mr. Goodeve's Elements of Mechanism.

on the generating circle, so that if we take a succession of points Q, Q1, Q2... on the generating circle, and describe with them a succession of epicycloids QP, Q1P1, Q2P2... they will all be exactly like one another, and if P'Q' be any epicycloid described on the same base with the same generating circle as the others, it too will be exactly like the rest: if we now suppose all the former to remain fixed, and the circle P'AC to revolve round its centre, carrying P'q' with it, then when p' comes to P2, the curve P'Q' will exactly cover P2 Q2, and in like manner it will successively cover P1 Q1 and PQ.

Proposition 22.

An epicycloidal tooth can be made to work correctly with a straight tooth.

Let PQ be the tooth described on the base a P, the centre

of which is o1, by a circle whose diameter is AO; suppose the base to revolve round o, and let the tooth assume successively the positions P11 P2929 P313...cutting the circle ADO in points 91, 92, 93, then since the straight lines 091, 092, 093.... touch the epicycloid in the points 91, 92, 93

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round o, will, if driven by the tooth, come successively

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into the positions o a1, ο α2, O A3 passing through the points 91, 92, 93 • respectively. Now, if we suppose the angles A O1P1, P112. to be equal, the arcs AP1, PIP2 P2P3・・・ are equal, and therefore (Art. 108 (a)) the arcs A 91, 91929 9293••• are equal, and the angles they subtend at the centre cwill be equal, and their halves will be also equal, i.e. the angles AoA1, A10A2, A2003, are equal; so that if the circle PAO, move with a uniform angular motion, it will communicate a uniform angular motion to a straight line ao movable about the point o, i.e. the straight line works truly with the epicycloidal tooth.

Ex. 543. If with centre o and radius On a circle be described, show that if this circle work with AP by friction, any one of its radii will have the same angular velocity as if it had been driven by the tooth PQ.

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FIG. 149.

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109. Practical Rule for the Form of Teeth.*-Let 0, 01 be the centres of the two toothed wheels; draw the line of centres 001; when the point of contact of any two teeth is on the line of centres let it be at A; with centres o and o and radii o A and o, a respectively describe circles, a aα', bab'; these are called the pitch circles of the respective wheels, i. e. the two circles which rolling by friction would move with the same angular motions as the wheels. Now, if there are to be m teeth in the wheel o, there must be m1 in the wheel 01, where m1 is given by the proportion o A : OLA :: m:m1.

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Divide the circumference of aaa' into m equal parts, * This rule, though not the best, is-or, at all events, used to be-very generally employed in practice. See Willis, p. 106.

of which parts let A A, be one; the chord of this are is called the pitch of the wheel; divide it into two (nearly)

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equal parts, of these AE (the smaller) is the breadth of a tooth, and EA, the space between two teeth; then the flanks BA, DE of a tooth (i.e. the parts of its outline within the pitch circle) are straight lines converging to the centre o; and the faces of the tooth AC, EF (i.e. the parts of its outline on the outside of the pitch circle) are portions of epicycloids described on the pitch circle as a base by a generating circle whose diameter equals the radius of the pitch circle of the wheel with which it is to work, viz. OLA. The teeth of the wheel o, are cut upon the same principle; the circumference of the pitch circle bab' is divided into m, equal parts, and each is divided into a tooth and a space; the flanks of the teeth converge to o1, the faces are epicycloids described on the pitch circle as a base by a generating circle whose diameter equals the radius O A. That the two wheels thus constructed will work truly, follows immediately from Prop. 22; thus, if the wheel o revolve uniformly, the tooth BAC driving the tooth B'AC', the epicycloid AC will cause the straight line A B,' and therefore the wheel o1, to revolve uniformly: on the other hand, if the wheel o, moving with a uniform motion drive o, the epicycloid AC' will cause the straight

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