Now suppose that on the cylindrical surface of a spur-wheel B c, ([fig. 3]) we cut oblique or rather screw-formed teeth, of which two are shewn at a c, b d, so inclined to the plane of the wheel, as that the end c of the tooth a c may not pass the plane of the axes A B c, until the end b of the other tooth b d has arrived at it, this wheel will virtually be divided into an infinite number of teeth, or at least into a number greater than that of the particles of matter, contained in a circular line of the wheel’s circumference. For suppose the surface of a similar, but longer cylinder, stripped from it and stretched on the plane A B C E ([fig. 4]) where the former oblique line will become the hypothenuse B C, of the right angled triangle C A B, and will represent all the teeth of the given wheel, according to the sketch E G at the bottom of the diagram. Here the lines A B and C E, are equal to the circumference of the base of the cylinder, and A C and B E to its length; and if between A and B, there exist a number, m, of particles of matter, and between A and C a number, n, the whole surfaced A B C E will contain m n particles, or the product of m and n; and the line B C, will contain a number = √m² + n², from a well known theorem; whence it appears that the line B C is necessarily longer than A B, and hence contains more particles of matter.[2]
[2] It need hardly be observed, that whatever is true of the whole triangle C A B, ([fig. 4]) is true of every similar part of it, be it ever so small: and in fact, when the hypothenuse B C, is folded again round the cylinder, from which we have supposed it stripped, the acting part will be very small indeed; but it will still act in the way here described, and give tendencies to the wheel it acts on, and to its axis, precisely proportionate to the quantities here mentioned.
It is besides evident, that the difference between the lines B C and A B, depends on the angle A C B; in the choice of which, there is a considerable latitude. For general use however, I have chosen an angle of obliquity of 15°, which I shall now assume as the basis of the following calculations. The tangent of 15°, per tables, is in round numbers 268 to radius 1000; and the object now is to find the number of particles in the oblique line B C, when the line A B, contains any other number, t.
By geometry, B C(x) = √r² + t² = √1000² + 268² = 1035 nearly; and this last number is to 268, as the number of particles in the oblique line B C is to the number contained in the circumference A B, of the base of the cylinder. Hence it appears, that a wheel cut into teeth of this form, contains (virtually) about four times as many teeth, as a wheel of the same diameter, but indefinitely thin, would contain. And the disproportion might be increased, by adopting a smaller angle.
Thus I apprehend it is proved, that the action of a wheel of this kind, on another with which it geers, is perfectly uniform in respect of swiftness; and hence the proof that it is likewise so, as to the force communicated.
Before I proceed to the second proposition, I ought perhaps to anticipate some objections that have been made to this system of geering, and which may have already occurred to some gentlemen present. For example, it has been supposed that the friction of these teeth, is augmented by their inclination to the plane of the wheel; but I dare presume to have already proved, that it is this very obliquity, joined to the total absence of motion in direction of the axes, that destroys the friction, instead of creating it. I acknowledge however, that the pressure on the points of contact, is greater than it would be on teeth, parallel to the axes of the wheels, and I farther concede that this pressure tends to displace the wheels in the direction of the axes, (unless this tendency is destroyed by a tooth, with two opposite inclinations.) But supposing this counteraction neglected, let us ascertain the importance of these objections. First, with regard to the increase of pressure on the point D of the line B C, (representing the oblique tooth in question,) relative to that which would be on the line B E, (which represents a tooth of common geering:) let A D be drawn perpendicular to B C. If the point D can slide freely on the line B C, (and this is the most favourable supposition for the objection,) its pressure will be exerted perpendicularly to this line; and if the point A, moves from A to B, the point D, leaving at the same moment the point A, and moving in direction A D, will only arrive at D in the same time, its motion having been slower than that of A, in the proportion of A B to A D; whence by the principle of virtual velocities, its pressure on B C is to that on A C, as the said lines A B to D A.
To convert these pressures into numbers, according to the above data; we have A C = 1000, A B = 268, B C = 1035; then from the similar triangles B A C, B D A, it will be B C : A C ∷ A B : A D = 268000⁄1035 = 259 nearly. Therefore the pressure on B C, is to that on A C, as 268 to 259, or as 1035 : 1000.