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DAI will make n, revolutions; so that whilst any other circle GH of the one cone makes n, revolutions, the corresponding circle HK of the other cone will make n, revolu tions: but n, times the circumference of the circle GH is equal to n, times that of the circle HK, for the diameters of these circles, and therefore their circumferences, are to one another (by similar triangles) in the same proportion as the diameters and the circumferences of the circles DP and DI. Since, then, whilst the cones make n, and n, révolutions respectively, the circles HG and HK are carried through n, and n, revolutions respectively, and that n, times the circumference of HG is equal to n, times that of HK, therefore the circles HG and HK roll in contact through the whole of that space, nowhere sliding upon one another. And the same is true of any other corresponding circles on the cones; whence it follows that their whole surfaces are made to roll upon one another by their mutual contact, no two parts being made to slide upon one another by the rolling of the

rest.

The rotation of the one axis might therefore be communicated to the other by the rolling of two such cones in contact, the surface of the one cone carrying with it the surface of the other, along the line of contact AD, by reason of the mutual friction of their surfaces, supposing that they could be so pressed upon one another as to produce a friction equal to the pressure under which the motion is communicated, or the work transferred. In such a case, the angular velocities of the two axes would evidently be to one another (equation 227) inversely, as the circumferences of any two corresponding circles DP and DI upon the cones, or inversely as their radii ND and OD, that is (by construction) inversely as the numbers and teeth which it is supposed to cut upon the wheels.

When, however, any considerable pressure accompanies the motion to be communicated, the friction of two such cones becomes insufficient, and it becomes necessary to transfer it by the intervention of bevil teeth. It is the characteristic property of these teeth that they cause the motion to be transferred by their successive contact, precisely as it would by the continued contact of the surfaces of the

cones.

229. To describe the teeth of bevil wheels.*

From D let FDE be wrawn at right angles to AD, intersecting the axes AE and AF of the two cones in E and F; suppose conical surfaces to be generated by the revolution of the lines DE and DF about AE and AF respectively;

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and let these conical surfaces be truncated by planes LM and XY respectively perpendicular to their axes AE and AF, leaving the distances DL and DY about equal to the depths which it is proposed to assign to the teeth. Let now the conical surface LDPM be conceived to be developed upon a plane perpendicular to AD, and passing through the point D, and let the conical surface XIDY be in like manner developed, and upon the same plane. When thus

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developed, these conical surfaces will have become the plane surfaces of two segmental annuli MPpm and IXxit, whose centres are in the points E and F of the axes AE and AF, and which touch one another in the point D of the line of contact AD of the cones.

Let now plane or spur teeth be struck upon the circles Pp and Ii, such as would cause them

* The method here given appears first to have been published by Mr. Tredgold in his edition of Buchanan's Essay on Millwork, 1823, p. 103.

The lines MP and pm in the development, coincided upon the cone, as also the lines IX and ix; the other letters upon the development in the above

to drive one another as they would be driven by their mutual contact; that is, let these circles Pp and I be taken as the pitch circles of such teeth, and let the teeth be described, by any of the methods before explained, so that they may drive one another correctly. Let, moreover, their pitches be such, that there may be placed as many such teeth on the circumference Pp as there are to be teeth upon the bevil wheel HP, and as many on I as upon the

wheel HI.

Having struck upon a flexible surface as many of the first teeth as are necessary to constitute a pattern, apply it to the conical surface DLMP, and trace off the teeth from it upon that surface, and proceed in the same manner with the surface DIXY.

Take DII equal to the proposed lengths of the teeth, draw ef through H perpendicular to AD, and terminate the wheels at their lesser extremities by concave surfaces HGml and HKey described in the same way as the convex surfaces which form their greater extremities. Proceed, moreover, in the construction of pattern teeth precisely in the same way in respect to those surfaces as the other; and trace out the teeth from these patterns on the lesser extremities as on the greater, taking care that any two similar points in the teeth traced upon the greater and lesser extremities shall lie in the same straight line passing through A. The pattern teeth thus traced upon the two extremities of the wheels are the extreme boundaries or edges of the teeth to be placed upon them, and are a sufficient guide to the workman in cutting them.

230. To prove that teeth thus constructed will work truly with one another.

It is evident that if two exceedingly thin wheels had been taken in a plane perpendicular to AD (fig. p. 286.) passing

figure represent points which are identical with those shown by the same letters in the preceding figure. In that figure the conical surfaces are shown developed, not in a plane perpendicular to AD, but in the plane which contains that line and the lines AE and AF, and which is perpendicular to the last-men tioned plane. It is evidently unnecessary, in the construction of the pattern teeth, actually to develope the conical extremities of the wheels as above described; we have only to determine the lengths of the radii DE and DF by construction, and with them to describe two ares, Pp, Ii, for the pitch circles of the teeth, and to set off the pitches upon them of the same lengths as the pitches upon the circles DP and DI, which last are determined by the numbers of teeth required to be cut upon the wheels respectively.

through the point D, and having their centres in E and F; and if teeth had been cut upon these wheels according to the pattern above described, then would these wheels have worked truly with one another, and the ratio of their angu lar velocities have been inversely that of ED to FD, or (by similar triangles) inversely that of ND to OD; which is the ratio required to be given to the angular velocities of the bevil wheels.

Now it is evident that that portion of each of the conical surfaces DPML and DIXY which is at any instant passing through the line LY is at that instant revolving in the plane perpendicular to AD which passes through the point D, the one surface revolving in that plane about the centre E, and the other about the centre F; those portions of the teeth of the bevil wheels which lie in these two conical surfaces will therefore drive one another truly, at the instant when they are passing through the line LY, if they be cut of the forms which they must have had to drive one another truly (and with the required ratio of their angular velocities) had they acted entirely in the above-mentioned plane perpendicular to AD and round the centres E and F. Now this is precisely the form in which they have been cut. Those portions of the bevil teeth which lie in the conical surfaces DPML and DIXY will therefore drive one another truly at the instant when they pass through the line LY; and therefore they will drive one another truly through an exceedingly small distance on either side of that line. Now it is only through an exceedingly small distance on either side of that line that any two given teeth remain in contact with one another. Thus, then, it follows that those portions of the teeth which lie in the conical surfaces DM and DX work truly with one another.

Now conceive the faces of the teeth to be intersected by an infinity of conical surfaces parallel and similar to DM and DX; precisely in the same way it may be shown that those portions of the teeth which lie in each of this infinite num ber of conical surfaces work truly with one another; whence it follows that the whole surfaces of the teeth, constructed as above, work truly together.

231. THE MODULUS OF A SYSTEM OF TWO CONICAL OR
BEVIL WHEELS.

Let the pressure P, and P, be applied to the conical

wheels represented in the accompanying figure at perpendicular distances a, and a, from their axes CB and CG; let the length AF of their teeth be represented by b; let the distance of any point in this line from F be represented by r, and conceive it to be divided into an exceedingly great number of equal parts, each represented by Ax. Through each of these points of division imagine planes to be drawn

K

I

perpendicular to the axes CB and CG of the wheels, dividing the whole of each wheel into elements or laminae of equal thickness; and let the pressures P, and P, be conceived to be equally distributed to these lamina. The pressure thus distributed to each will then be represented by Ax on the

P

P,

one wheel, and Ax on the other. Let p, and p, represent the two pressures thus applied to the extreme laminæ AH and AK of the wheels, and let them be in equilibrium when thus applied to those sections separately and independently of the rest; then if R represent the pressure sustained along that narrow portion of the surface of contact of the teeth of the wheels which is included within these laminæ, and if R, and R, represent the resolved parts of the pressure R in the directions of the planes AH and AK of these laminæ, the pressures p, and R, applied to the circle AH are pressures in equilibrium, as also the pressures p, and R, applied to the circle AK. If, therefore, we represent as before (Art. 216.) by m, and m,, the perpendiculars from B and G upon the

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