action of the teeth in effecting a constant velocity ratio will not be impaired. A back-lash will therefore be introduced, which will be the greater the more the wheels are withdrawn from each other. In any given pair of involute wheels, therefore, we can, by properly adjusting by trial the distance of their centers, reduce the back-lash to the least quantity that will allow the teeth to act without jamming. This advantage is possessed by no other form, and particularly recommends these teeth for dial-work, or any such kinds of mechanism, in which the back-lash is mischievous.

129. To apply involutes to rack-teeth.

Describe a pitch circle, (fig. 57,) radius BT, and draw AC a tangent at T for a pitch line to the rack; let the

57

E

m

G

H

circle whose radius is BE be the base of an involute EF, and let the tooth of the rack be bounded by a straight line EGH, making an angle EGA with the pitch line equal to BTE. If the involute be moved to ef, it will drive the sloped tooth to gh, always touching it in the line ETh; and the velocity of the circumference of the pitch circle will always equal that of the pitch line: for

also Eh arc Ee, by the property of the involute

= arc mn x sin BTE;

= arc mn × sin EGT; .. Gg = arc mn.

This may be shewn from fig. 56, page 113. For let the radius of the wheel AT become infinite, then will the pitch line be a straight line passing through T, and touching the pitch

58

circle of the wheel whose center is B, and the involutes GH, Em will become right lines perpendicular to the line ETD. Thus is obtained a rack with straight-sided sloping teeth, as in fig. 58.

Hence a wheel with involute teeth will work with a rack whose teeth are straight-sided and inclined to the pitch line at an angle 0, provided

In such a rack, the locus of contact being the tangent line ETh, the contact will not be confined to a single point of the tooth, as it is in the common involute rack teeth, (Art. 106) which are derived from that particular case of this figure, in which the radius of the base coinciding with that of the pitch circle, the line ETh coincides with the pitch line of the rack. But a rack with sloping teeth will be pressed downwards by a resolved portion of the working pressure,

and this appears to me to be in many cases advantageous, and destructive of vibration.

To approximate to the true form of a tooth by arcs of circles.

130. The portion of curve employed in a tooth is so short, that a circular arc might be substituted for it with sufficient accuracy for all practical purposes, if its center and radius were determined upon correct principles.

In fact, practically the edges of teeth are always made ares of circles, but unfortunately, these arcs are often struck from the merest empirical rules, such as setting the point of the compasses in the pitch line on one side of the tooth, in order to strike the other, and vice versa, or similar absurdities*. Teeth have even been set out by forming their edges into semicircles struck alternately without and within the pitch circle; these are technically known by the name of hollows and rounds.

Some millwrights, with equal neglect of principle, give their teeth plane faces passing through the axis of the wheel, expecting them to wear themselves in a short time into proper forms. But the best workmen endeavour to give to their wheels teeth of the epicycloidal form, according to the rules laid down in Camust, or in Buchanan's Treatise on Millwork, which are immediately derived from Camus. In truth, the question is one of great practical importance; I do not mean to say, that it is necessary, or even practicable, to shape the teeth of small wheels into exact epicycloids or involutes, such as those which have been described in the preceding pages; but I do assert, that unless the rules for shaping them be derived from such considerations, so as to approximate their form to the true ones, as nearly as

* Vide Imison's School of Arts, or Gray's Experienced Millwright. + Camus on the Teeth of Wheels, 1806 and 1837.

1808, 1823 and 1841.

possible, that the action of the machines will be irregular and noisy, producing those vibrations which must be familiar to all who have been in the habit of examining machinery, and which are above all things conducive to the wearing out and disintegration of every part of the mechanism. The investigation of the proper curves for the teeth of wheels is, therefore, by no means one of mere curiosity, although this has been sometimes hastily asserted. One proof of the necessity of attending to the exact theoretical forms, is the acknowledged impossibility of making one wheel to work with two others whose numbers of teeth are different, by means of the usual rules.

131. The method employed by the best workmen for shaping the teeth of a proposed wheel, or of a pattern from which to cast one, is as follows:

The shape of a single tooth adapted for this wheel is traced in the true epicycloidal form, by means of templets, that is, of a pair of boards whose edges are cut to the curvature of the pitch circle, and describing circle respectively, and which may be termed the pitch templet and the describing templet. The latter carries a describing point in its circumference, and by rolling its edge upon that of the pitch templet, the arc required for the face of the tooth is traced upon the drawing board *.

This done, the workman finds with his compasses, by trial, a center and small radius, by which an arc of a circle can be described, that will coincide as nearly as he can manage to make it with the templet-traced epicycloid.

If the method I have recommended under the third solution (Art. 114) be adopted, then one describing templet will serve for the entire set; but since this templet is required to trace hypocycloids for the flanks, as well as epicycloids for the faces, every pitch templet must have a convex and a concave edge, both shaped into an arc of the pitch circle of the wheel in question. The concave edge is not required upon the common system (Art. 98), because the flanks are radial lines.

Then, having struck upon the fronts of the rough cogs a circle which is concentric with the pitch circle, and whose distance from it is equal to that of the center of his small arc, he adjusts his compasses to the small radius, and always keeping one point in the circle just described, he steps with the other to each cog in succession, they having been previously divided into equal parts corresponding to the given pitch and breadth of the teeth; upon each cog he describes two arcs, one to the right and one to the left, which serve him as guides in shaping and finishing the acting faces.

132. The practical convenience of this method is very great, and appears to require only a more commodious and certain method of determining the center and radius of the approximate arc.

The first method that suggests itself, is to find the center and radius of the circle of curvature at some intermediate point between the extremities of the curve selected for the teeth, and to substitute an arc of this circle in lieu of the actual curve. But the determination of the required circles may be effected upon general principles, without taking individual curves into the considerations. In fact, Euler, in his elaborate paper on the Teeth of Wheels*, undertook to investigate a general expression for curves that possess the property of revolving in contact with a constant velocity ratio, which he effected by determining the relation between their radii of curvature; and suggested that in practice small arcs of the circles of curvature thus obtained would probably suffice for the sides of teeth. He accordingly gave some geometrical constructions for this purpose, but the hint thus supplied was neglected by every subsequent writer, partly, perhaps, by reason of the abstruse manner in which he treated the subject.

N. C. Pet. XI. 209.