In well-constructed machinery there should never be fewer than seven contacts in the system, since of these only four are working; and therefore only two teeth are fully engaged; and it is necessary that two teeth be engaged at once, in order that the wearing may tend to remove any unavoidable inequalities of workmanship.
When we attempt to delineate the forms of wheels with few teeth by help of any of these orbits, we find that the contours overlap each other; in such cases the following tooth of the conjugate wheel effaces, as it were, the trace belonging to the preceding tooth; and the contacts, though still holding good of the geo metrical curves, become mechanically impossible.
Thus it is that there are lintits below which we cannot go in the numbers of the teeth. If the overlapping occur at the shoulder of the tooth some of the useful contacts are wanting; but when the replication is only at the point of the tooth, the want of the supplemen tary contact, occasions no inconvenience. An examin ation of the different cases shows that, with seven contacts, the smallest numbers which can be used on the three systems just mentioned are 19, 17, and 11, so that the system of epicycloidal teeth has, in this respect, the advantage over the others. Clock pinions, then, should not have fewer than eleven leaves. This method of considering the subject was first published by the writer of this article in A New General Theory of the Teeth of Wheels (Edinburgh, 1852).
It remains to cut the actual wheels to the shapes thus formed. The essentials of the operation are these: .The blank wheel is attached to the axis of a large divided cir cle, which can be turned round and held in any desired position. A. cutter, generally a revolving cutter, is brought down upon the blank, so as to notch out the space between two teeth; this done, the circle is turned round by the proper number of divisions, and snother space is cut, and in this way the whole circumference of the wheel is gone over. In order that the work be well done, it is essential that the cutter be truly shaped; and when the edges get blunted by use, it is no easy matter to avoid spoiling the shape in the resharpening. Whatever system be followed, the form of the tooth varies from one number to another, so that the. cutter which answers for a wheel of 20 cannot do for one of 30 teeth; and hence, when accurate results are wanted, there must be a cutter for each wheel. In order to avoid the expense of so many cutters, each requiring to be
carefully made, the slovenly practice is too often followed of having, perhaps, two cut ters, one to be used for pinions, the other for wheels; and the result is the intolerable noise which is so common in mills, and which, if properly understood, should be taken as an indication of unnecessary expenditure of power.
When, min the wholesale manufacture of clocks and watches, multitudes of wheel% are to be cut of one size, careful attention can be given to the shape of the cutter. The labor is economized by binding a considerable number of blanks together on the divid ing engine, and plowing out the teeth of the whole of them at once. For the small wheels, technically called pinions, which cannot conveniently be fixed on the dividing-engine, pinion,-wires are used; these are wires of brass or steel drawn through holes of the proper shape, and having the leaves running all along them. The watchmaker removes the leaves from those parts ;where they are not wanted, and thus obtains the pinion and its axle in one piece; in this way he gains the advantages of solidity and economy of work mansittp.
Among the many purposes to which wheel-work is applied, it sometimes happens i that an unequable motion is wanted. Thus, in the construction of an orrery, it is desira ble that while one index turns uniformly to show the time, another may turn so as to show the unequal motion of the sun in the ecliptic. In that case the variations of the i velocity are small, and it is enough to divide the teeth unequally, as the slight nequal ity can hardly affect the working of the apparatus. But when the changes of velocity are considerable, the matter must be more carefully looked into. If we suppose the pitch-lines of two wheels to be uneven, and to roll upon each other without regard to the positions of their centers, the forms of teeth to be arranged upon those pitch-lines may be traced out almost in the manner already explained for round wheels. The pitch line must be divided into equal distances, and the disk must receive a half-slidingi half turning motion, so that the pitch-line may pass through the point C (fig. 4) always per pendicularly to the line AB, which is the line of centers for round wheels. The combi nation of this motion with the proper motion of the points of contact gives true forms for The teeth.