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be acting at the same time, (twice the number that can be admitted in M. De la Hire's method.) This, by dividing the pressure among several teeth, diminishes its quantity on any one of them, and therefore diminishes the dents or impressions which they unavoidably make on each other. It is not altogether free from sliding and friction, but the whole of it can hardly be said to be sensible. The whole slide of a tooth, three inches long, belonging to a wheel of ten feet diameter, does not amount to th of an inch, a quantity altogether insignificant.

In the same article, this highly respectable philosopher was mistaken, in supposing, with other eminent authors, that the mutual action of the teeth, (when formed into epicycloids, by the method of M. Camus,) is absolutely without friction, and in saying, That one tooth only APPLIES itself to the other, and ROLLS on it, but does not SLIDE or RUB on it in the smallest

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degree. This makes them last long, or rather, does not allow them to wear." A very slight examination of the figures given in various parts of the preceding Essay, will, I hope, show, that the point of contact must slide from the pitch line of the conducting tooth outwards. Dr. Young, in his Natural Philosophy, Vol. II. page 183, says, that "a form [of teeth,] without friction, is perfectly impracticable, although, for a single tooth, possible."

In the first volume of the same work, he makes the following judicious observations on our present subject:

"It has been supposed by some of the best authors that the epicycloidal tooth has also the advantage of completely avoiding friction; this is however by no means true, and it is even impracticable to invent any form for the teeth of a wheel, which will enable them to act on other teeth without friction.

"In order to diminish it as much as possible, the teeth must be as small and as numerous as is consistent with strength and durability; for the effect of friction always increases with the distance of the point of contact from the line joining the centres of the wheels.

"In calculating the quantity of the friction, the velocity with which the parts slide over each other has generally been taken for its measure: this is a slight inaccuracy of conception, for, as we have already seen, the actual resistance is not at all increased by increasing the relative velocity; but the effect of that resistance, in retarding the motion of the wheels, may be shown, from the general laws of mechanics, to be proportional to the relative velocity thus ascertained. When it is possible to make one wheel act on teeth fixed in the concave surface of another, the friction may be thus diminished in the proportion of the difference of the diameters to their sum.

"If the face of the teeth, where they are in contact, is too much inclined to the radius, their mutual friction is not much affected, but a great pressure on their axes is produced; and this occasions a strain on the machinery, as well as an increase of the friction on the axes."*

The concluding part of these observations appears to me peculiarly applicable to the figure of teeth described in our last chapter; for in wearing, they will be more liable than many other forms, to have the face of the teeth, where they are in contact,

too much inclined to the radius.

* Young's Lectures, Vol. I. p. 176.

Remarks on the Friction of Wheel Work, and on the Forms best suited for Teeth.

In a Letter from DR. YOUNG.

"I have been considering your observations on the difference of the friction, accordingly as the teeth touch before or after the line of centres; at first I was disposed to doubt of the fact; but upon more mature examination, I found that, like many other practical observations, they went beyond the scope of the doctrines of theoretical writers. I cannot however perfectly agree with you as to the explanation of the fact; but I will state to you briefly my opinion on the subject, not having leisure at present to enter into a more ample discussion.'

"The magnitude of the friction has usually been estimated by the relative velocity of the surfaces concerned; a mode of calculation, which, as I have

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