part of the pitch; but the least necessary length may be found experimentally by replacing the templet в on the board F, and making p coincide with a, roll c towards E till it touches B in b, the corresponding face of the next tooth; mark then the position of the tracing point, and through this point draw an arc from the centre g of the wheel. This arc will mark the extremity of the tooth, and the arc g p will be the true radius of the wheel.

This process, which, though complicated in description, is very easy in practice, must be repeated with two templets cut to the pitch circle of the pinion, the same generating circle c being employed; a similar pattern tooth will thus be found for the pinion, which will work with that already found for the wheel. The usual custom in practice is for the millwright first to describe the epicycloidal and hypocycloidal forms of the teeth required in the wheel and pinion; he then constructs two model teeth, one for the wheel and the other for the pinion, and from these he determines the true curves, and by means of his compasses transfers the same to the wheels or patterns on which these forms are to be impressed. The generating circle, it may be observed, must not exceed in size the radius of the pinion, or it would give rise to a weak form of tooth, thinner at the root than at the pitch circle.

Second method, where two generating circles are employed, in order that the flanks of the teeth may be straight lines radii of the wheel and pinion respectively.

It is the usual practice of millwrights to make the parts of the teeth of wheels within the pitch circles radii of the wheel. Now, we have seen that a hypocycloid described by a generating circle equal in diameter to the radius of the wheel would be a diameter of the wheel. If, therefore, the flank of the tooth of the wheel and the face of the tooth of the pinion be described by a templet cut to a radius equal to half that of the wheel, and the flank of the tooth of the pinion and face of that of the wheel be described by a templet cut to a radius equal to half that of the pinion, then these teeth will work together truly and will have radial flanks.

Since it is unnecessary to describe the flanks of such teeth by templets, there will be needed only one templet cut to the pitch circle of each wheel, but templets of two generating circles are required. In other respects the method is identical with

that already described. The great defect of this method is, that neither the wheel nor pinion will work accurately with a wheel or pinion of any other diameter than that for which they were originally made, and thus a vast number of wheel patterns must be made to fulfil the requirements of practice; whereas wheels described by the previous method will work equally well with all other wheels the teeth of which have been described by the same generating circle--it being understood that only the parts of teeth without the pitch circle of the wheel roll on the parts within the pitch circle of the pinion, and those without the pitch circle of the pinion on those within the pitch circle of the wheel.

Hence Professor Willis has been led to suggest that for a given set of wheels a constant generating circle should be taken to describe both the parts without and within the pitch circles of the whole series, instead of making that circle depend on the diameters of the wheels. In this case the first solution must be employed, and the flanks of the teeth will not be straight; but the great advantage is gained, that any pair of wheels in the series will work together equally well.

To determine the proper size of the generating circle, we must remember that a tooth of weak form is produced when the generating circle is greater than half the diameter of the wheel. Hence the generating circle may be best made of a diameter equal to the radius of the smallest pinion of the series which are to work together.

The Rack is the extreme case of a wheel, or may be considered as a wheel of infinite radius. It may be described by either of the methods above; only noting that, if the second method be employed, the generating circle which traces the face of the teeth of the wheel becomes a straight line, and the epicycloid becomes an involute.

If the teeth of a series of wheels and of a rack be described by the same generating circle, any of the wheels will work with equal accuracy into the rack.

Involute Teeth.

The Involute, the curve traced by a flexible line unwinding from the circumference of a circle, is called an involute.

Let p and w (fig. 194) be the pitch lines of a wheel and

pinion, and let A and B be their centres. From A and B describe two circles D C, with radii a b and в b of the wheel and pinion respectively; so that

AC BC AD BC

Let m n and op be two involute curves described by flexible lines unrolling from the circles D and c respectively, and touching at b. Then if b c, b D be drawn tangents to the circles at the points D and c, they are also in one straight line, because they are both normals to the

curves at b. It may also be shown that the line C D intersects A B in c where the pitch lines touch. Hence we have found two curves such, that the line perpendicular to their common tangent passes in all positions of the wheel and pinion through c, which is the sufficient condition of their uniform motion, if moved by the sliding of the curves instead of by contact at c. Hence, if the

wheels be constructed with teeth formed to these involute curves, they will work with perfect regularity of motion.

P

Fig. 194.

B

D

In practice, the chief condition to be observed is to diminish the pressure on the axes, which is the chief defect of this form of teeth. The common tangent should be drawn through c, making an angle with A B, not deviating more than 20° from a right angle. Involute wheels have the double advantage that they work equally well if, through the wear of the brasses, the wheels have receded from one another; and any involute wheels of the same pitch and similarly described-that is, having the common tangent to the base circles passing through the point of contact of the pitch lines; or, in other words, base circles proportional to the primitive radii-will work together.

Mr. Hawkins, the translator of Camus' first proposed a simple instrument for describing the teeth of wheels to an invo

lute curve. It consists of a straight piece of watch-spring a b (fig. 195), with a screw at one end, and filed away at the edges so as to leave two teeth or tracers, c c, projecting from the edges of the watch-spring. At b a bit of wire is put through and riveted, so as to form a knot by which the spring can be firmly held and stretched, as it is unwound from the base on which the involute is generated. This watch-spring is screwed to the edge of a templet A, curved to the radius of the base circle of the involute: and this being placed so that its centre coincides with the centre of the wheel, and revolved to bring one of the tracing points c in succession to each of the points at which corresponding faces of the teeth cut the pitch line, a series of involute curves may be described by unfolding the watch-spring, whilst keeping it firmly stretched tangentially to the sector to which it is fixed. The sector a must then be

turned over, and the involutes of the opposite faces of the teeth struck in a similar manner.

Another plan is to employ a straight ruler instead of the watch-spring, a tracer being fixed in its edge. This shows that the involute is an epicycloid generated by a straight line. The ruler must be kept in contact with the base circle, and the tracer brought in succession to all the points in which the faces of the teeth cut the pitch line.

Hence, to describe a wheel with involute teeth, the line of centres must be drawn and divided proportionally to the number of teeth in the wheel and pinion. Draw the pitch line; divide the pitch line into the same number of equal parts as there are teeth in the wheel, and at these points mark out the thicknesses

of the teeth all round. Draw the tangent to the base circles, making an angle of about 80° with the line of centres, which will give the radius of the base circle drawn touching it. A templet must be made to this radius, and then the involutes may be drawn by either of the preceding methods.

Allowance must be made to permit free play of the teeth in the spaces, the teeth being somewhat shorter than the distance between the bases of the involutes. But wheels of this figure require but little play in the engagement.

In the case of racks, the rack teeth are bounded by straight lines perpendicular to the tangent drawn from the point where the pitch lines touch, to the base circle from which the involutes of the wheel are struck. If the teeth of the rack be made rectangular-that is, bounded by lines perpendicular to the pitch line-the involute must be struck from a base circle equal to the pitch circle of the wheel. In the former case there is a downward pressure on the rack; in the latter, the teeth of the wheel touch those of the rack in a single point-namely, the pitch line of the latter.

Professor Willis's Method of Striking the Teeth of Wheels.

In practice, the custom of describing the teeth of wheels as arcs of circles, has, from its simplicity, been generally adopted. The methods already given, however simple, when adopted in the formation of a single tooth, become tedious in their applications to wheels of large size; and to this must be added the imperfect comprehension of their advantages by the millwrights charged with the task of designing wheel patterns.

Circular arcs struck at random, according to the judgment of the millwright, are often employed; and even where better principles have been introduced, it is common, after describing a single tooth accurately, to find by trial a circular are nearly corresponding with its curve, and to employ this in marking out the cogs of the required wheel.

Seeing the advantages of the circular arc, and believing that it is not objectionable if only the employment of it is guided by true principles, Professor Willis has rendered this great service to practical mechanics--he has shown how, by a simple construction, the arcs of circles may be found, which, used in the construction of the teeth of wheels, will work truly on each other.