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work with any other. This offers, under many circumstances great advantages, especially in the very great reduction of the number of patterns which he will be required to keep. There are, moreover, many cases in which some arrange ment similar to this is indispensable to the true working of the wheels, as when one wheel is required (which is often the case) to work with two or three others, of different numbers of teeth, A for instance to turn B and C; by the ordinary method of construction this combination would be impracticable, so that the wheels should work truly. Any generating circle common to a whole set of the same pitch, satisfying the above condition, it may be asked whether there is any other consideration determining the best dimensions of this circle. There is such a consideration arising out of a limitation of the dimensions of the generating circle of the hypocycloidal portion of the tooth to a diameter not greater than half that of its base. As long as it remains within these limits, the hypocycloidal generated by it is of that concave form by which the flank of the tooth is made to spread itself, and the base of the tooth to widen; when it exceeds these limits, the flank of the tooth takes the convex form, the base of the tooth is thus contracted, and its strength diminished. Since then, the generating circle should not have a diameter greater than half that of any of the wheels of the set for which it is used, it will manifestly be the greatest which will satisfy this condition when its diameter is equal to half that of the least wheel of the set. Now no pinion should have less than twelve or fourteen teeth. Half the diameter of a wheel of the proposed pitch, which has twelve or fourteen teeth, is then the true diameter or the generating circle of the set. The above suggestions are due to Professor Willis.*

* Professor Willis has suggested a new and very ingenious method of striking the teeth of wheels by means of circular arcs. A detailed description of this method has been given by him in the Transactions of the Institution of Civil Engineers, vol. ii., accompanied by tables, &c., which render its prac tical application exceedingly simple and easy.

211. TO DESCRIBE INVOLUTE TEETH.

Let AD and AG represent the pitch circles of two wheels intended to work together. Draw a straight line FE through the point of contact A of the pitch circles and inclined to the line of centres CAB of these wheels at a certain angle FAC, the influence of the dimensions of which on the action of the teeth will hereafter be explained, but which appears usually to be taken not less than 80°.* Describe two circles eEK and fFL from the centres B and C, each touching the straight line EF. These circles are to be taken as the buses from which the involute faces of the teeth are to be struck. It is evident (by the similar triangles ACF and AEB) that their radii CF and BE will be to one another as the radii CA and BA of the pitch circles, so that the condition necessary (Art. 201.) to the correct action of the teeth of the wheels will be satisfied, provided their faces be involutes to

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these two circles. Let AG and AH in the above figure represent arcs of the pitch circles of the wheels on an enlarged scale, and E, fL, corresponding portions of the circles eEK and fFL of the preceding figure. Also let Aa represent the pitch of one of the teeth of either wheel. Through the points A and a describe involutes ef and mu.†

* See Camus on the Teeth of Wheels, by Hawkins, p. 168.

Mr. Hawkins recommends the following as a convenient method of striking involute teeth, in his edition of "Camus on the Teeth of Wheels," p. 166. Take a thin board, or a plate of metal, and reduce its edge MN so as accurately to

Let be the point where the line EF intersects the involute mn; then if the teeth on the two wheels are to be nearly of the same thickness at their bases, bisect the line Ab in c; or if they are to be of different thicknesses, divide the line Ab in e in the same proportion*, and strike through the point c an involute curve hg, similar to ef, but inclined in the opposite direction. If the extremity fg of the tooth be then cut off so that it may just clear the circumference of the circle fL, the true form of the pattern involute tooth will be obtained.

There are two remarkable properties of involute teeth, by the combination of which they are distinguished from teeth of all other forms, and cæteris paribus rendered greatly preferable to all others. The first of these is, that any two wheels having teeth of the involute form, and of the same pitch, will work correctly together, since the forms of the teeth on any one such wheel are entirely independent of those on the wheel which is destined to work with it (Art. 201.) Any two wheels with involute teeth so made to work together will revolve precisely as they would by the actual contact of two circles, whose radii may be found by dividing the line joining their centres in the proportion of the radii of the generating circles of the involutes. This property involute teeth possess, however, in common with the epicycloidal teeth of different wheels, all of which are struck with the same generating circle (Art. 210.) The second no less important property of involute teeth-a property which distinguishes them from teeth of all other forms-is this, that they work equally well, however far the centres of the

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coincide with the circular are eE, and let a piece of thin watch-spring OR, having two projecting points upon it as shown at P, and which is of a width equal to the thickness of the plate, be fixed upon its edge by means of a screw 0. Let the edge of the plate be then made to coincide with the arc E in such a position that, when the spring is stretched, the point P in it may coincide with the point from which the tooth is to be struck; and the spring being kept continually stretched, and wound or unwound from the circle, the involute are is thus to be described by the point P upon the face of the board from which the pattern is to be cut.

*This rule is given by Mr. Hawkins (p. 170.); it can only be an approxima tion, but may be sufficiently near to the truth for practical purposes. It is to be observed that the teeth may have their bases in any other circles than those, fL and E, from which the involutes are struck.

The teeth being also of equal thicknesses at their bases, the method of ensuring which condition has been explained above.

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wheels are removed asunder from one another; so that the action of the teeth of two wheels is not impaired when their axes are displaced by that wearing of their brasses or collars, which soon results from a continned and a considerable strain. The existence of this property will readily be admitted, if we conceive AG and BI to represent the generating circles or bases of the teeth, and these to be placed with their centres C, and C, any distance. asunder, a band AB (p. 235., note) passing round both, and a point T in this band generating a curve mn, m'n' on the plane of each of the circles as they are made to revolve under it. It has been shown that these curves mn and m'n' will represent the faces of two teeth which will work truly with one another; moreover, that these curves are respectively involutes of the two circles AG and BHI, and are therefore wholly independent in respect to their forms of the distances of the centres of the circles from one another, depending only on the dimensions of the circles. Since then the circles would drive at any distance correctly by means of the band; since, moreover, at every such distance they would be driven by the curves mn and m'n' precisely as by the band; and since these curves would in every such position be the same curves, viz. involutes of the two circles, it follows that the same involute curves mn and m'n' would drive the circles correctly at whatever distances their centres were placed; and, therefore, that involute teeth would drive these wheels correctly at whatever distances the axes of those wheels were placed.

THE TEETH OF A RACK AND PINION.

212. To determine the pitch circle of the pinion. Let H represent the distance through which the rack is to be moved by each tooth of the pinion, and let these teeth be N in number; then will the rack be moved through the space N. II during one complete revolution of the wheel. Now the rack and pinion are to be driven by the action of their teeth, as they would by the contact of the circum

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213. To describe the teeth of the pinion, those of the rack being straight. The properties which have been shown to belong to involute teeth (Art. 201.) manifestly obtain, however great may be the dimensions of the pitch circle of their wheels, or whatever disproportion may exist between them. Of two wheels OF and OE with involute teeth which work together, let then the radius of the pitch circle of one OF become infinite, its circumference will then become a straight line represented by the face of a rack. Whilst the radius C,O of the pitch circle OF thus becomes infinite, that C,B of the circle from which its involute teeth are struck (bearing a constant ratio to the first) will also become infinite, so that the involute m'n' will become a straight line* perpendicular to the line AB given in position. The involute teeth on the wheel OF will thus become straight teeth (see fig. 1.), having their faces perpendicular to the line AB determined by drawing through the point O a tangent to the circle AC, from which the involute teeth of the pinion are struck. If the circle AC from which the involute teeth of the pinion. are struck coincide with its pitch circle, the line AB becomes

For it is evident that the extremity of a line of infinite length unwinding itself from the circumference of a circle of infinite diameter will describe, through a finite space, a straight line perpendicular to the circumference of the circle. The idea of giving an oblique position to the straight faces of the teeth of a rack appears first to have occurred to Professor Willis.

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