tooth; and if a circle be struck from the centre of the pitch circle passing through that point, all that portion of the tooth which lies beyond this circle may be cut off.*

The length of the tooth on the wheel intended to act with this, may be determined in like manner.

210. In the preceding article we have supposed the same generating circle to be used in striking the entire surfaces of the teeth on both wheels. It is not however necessary to the correct working of the teeth, that the same circle should thus be used in striking the entire surfaces of two teeth which act together, but only that the generating circle of every two portions of the two teeth which come into actual contact should be the same. Thus the flank of the driving tooth and the face of the driven tooth being in contact at

P in the accompanying figuret, this face of the one tooth and flank of the other must be respectively an epicycloid and a hypocycloid struck with the same generating circle. Again, the face of a driving tooth and the flank of a driven tooth being in contact at Q, these, too, must be struck by the

*The point e thus determined will, in some cases, fall beyond the extremity E of the tooth. In such cases it is therefore impossible to cut the tooth of such a length as to satisfy the required conditions, viz. that it shall drive only after it has passed the line of centres. A full discussion of these impossible cases will be found in Professor Willis's work (Arts. 102-104.).

†The upper wheel is here supposed to drive the lower.

same generating circle.

But it is evidently unnecessary that the generating circle used in the second case should be the same as that used in the first. Any generating circle will satisfy the conditions in either case (Art. 202.), provided it be the same for the epicycloid as for the hypocycloid which is to act with it.

According to a general (almost a universal) custom among mechanics, two different generating circles are thus used for striking the teeth on two wheels which are to act together, the diameter of the generating circle for striking the faces of the teeth on the one wheel being equal to the radius of the pitch circle of the other wheel. Thus if we call the wheels A and B, then the epicycloidal faces of the teeth on A, and the corresponding hypocycloidal flanks on B, are generated by a circle whose diameter is equal to the radius of the pitch circle of B. The hypocycloidal flanks of the teeth on B thus become straight lines (Art. 203.), whose directions are those of radii of that wheel. In like manner the epicycloidal faces of the teeth on B, and the corresponding hypocycloidal flanks of the teeth on A, are struck by a circle whose diameter is equal to the radius of the pitch circle of A; so that the hypocycloidal flanks of the teeth of A become in like manner straight lines, whose directions are those of radii of the wheel A. By this expedient of using two different generating circles, the flanks of the teeth on both wheels become straight lines, and the faces only are curved. The teeth shown in the above figure are of this form. The motive for giving this particular value to the generating circle appears to be no other than that saving of trouble which is offered by the substitution of a straight for a curved flank of the tooth. A more careful consideration of the subject, however, shows that there is no real economy of labour in this. In the first place, it renders necessary the use of two different generating circles or templets for striking the teeth of any given wheel or pinion, the curved portions of the teeth of the wheel being struck with a circle whose diameter equals half the diameter of the pinion, and the curved portions of the teeth of the pinion with a circle whose

T

diameter equals half that of the wheel. Now, one generating circle would have done for both, had the workman been contented to make the flanks of his teeth of the hypocycloidal forms corresponding to it. But there is yet a greater practical inconvenience in this method. A wheel and pinion thus constructed will only work with one another; neither will work truly any third wheel or pinion of a different number of teeth, although it have the same pitch. Thus the wheels A and B having each a given number of teeth, and being made to work with one another, will neither of them work truly with C of a different number of teeth of the same pitch. For that A may work truly with C, the face of its teeth must be struck with a generating circle, whose diameter is half that of C: but they are struck with a circle whose diameter is half that of B; the condition of uniform action is not therefore satisfied. Now let us suppose that the epicycloidal faces, and the hypocycloidal flanks of all the teeth A, B, and C had been struck with the same generating circle, and that all three had been of the same pitch, it is clear that any one of them would then have worked truly with any other, and that this would have been equally true of any number of teeth of the same pitch. Thus, then, the machinist may, by the use of the same generating circle, for all his pattern wheels of the same pitch, so construct them, as that any one wheel of that pitch shall 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 arrangement 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.*

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 ƒFL from the centres B

D

* 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 practical application exceedingly simple and easy.

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

and C, each touching the straight line EF. These circles are to be taken as the bases 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

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 eE, 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 mn.* Let b

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 coincide with the circular arc 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 O. Let the edge of the plate be then made to coincide with the arc eE 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 arc is thus to be described by the point P upon the face of the board from which the pattern is to be cut.