and journals. These defects can only be avoided by reducing, as far as praticable, the size of the teeth, and by the adoption of true principles in setting out their curvature in the original model.

To the first cause alone a large part of the perfect action of modern machinery of transmission is to be attributed; but there is moreover no doubt that, in practice, even where true principles have not been adopted, a considerable approach has been made to such forms as theory requires. Now, with certain limitations, it is known that if any form of tooth be taken for one wheel, there can be found another tooth which will work correctly with it. But there are certain forms which, being susceptible of accurate mathematical determination, are most convenient for the purpose. Camus, in 1752, was the first to work out the properties of epicycloidal and hypocycloidal curves when employed in the construction of the teeth of spur and bevel gearing. De la Hire adopted the same form. Euler, in 1760, and Kaestner, in 1771, investigated in a similar manner the properties of the involute. Since their time, Ferguson, Buchanan, Hawkins, Rennie, and Airy, have all contributed to perfecting the mathematical theory. And Professor Willis, amongst other important additions, has shown how a close approximation to a true form may be made by the adoption of a system of circular arcs.

From 1788, when Rennie completed the Albion

Mills, to the present time, wood and iron gear have been in general use for high velocities, and for every description of machinery where smoothness and accuracy of motion were required. Mr. Rennie was the first to introduce this system; and in most cases he made the driver, or large wheel, with wood cogs, and the driven, or pinion, of iron "chipped and turned "-that is, every tooth of the iron wheel was carefully divided in the pitch, having first been turned on the fane and the ends. of the teeth, and drawn to the epicycloidal form. They were then chipped with the hammer and chisel, and accurately filed to the required dimensions and forms. The same process was applied to the wooden teeth; and these wheels, when duly prepared, were keyed on their respective shafts, and securely fixed in contact in the mill.

The chipping and filing process has of late years been superseded by a cutting machine, which effects the same purpose, with less risk of error; and the good old system of a penny an inch, as practised in Rennie's time, has been exploded, much to the discomfiture of the old millwrights, who adhere with great tenacity to the hammer and chisel. Fig. 89 shows the cutting machine as constructed by Messrs. Peter Fairbairn and Co., of Leeds.

The object of this machine is not only to pitch and trim the teeth of a large spur or other wheel,

but to turn the face and sides of the segments previously, when bolted to the arms.

When used as a lathe for turning, the parts in use are as follows: B is a large headstock, carrying a hollow spindle (C), through which is inserted a mandrill upon which the wheel to be cut and turned is keyed. Provision is made for carrying the other end of this mandrill by a loose fixing. The hollow spindle is driven (with the wheel upon it) by a worm wheel (J) which is made to run loose on the spindle, but which is now by a lock bolt connected to the larger worm wheel or dividing wheel (E), the worm of which is now thrown out, and which is keyed firmly on the spindle. The necessary speeds are given by the five-speed cone and mitre gear. The tool for turning is held in an ordinary slide rest, which moves transversally on a saddle, which slides and is fastened in the T groves of two strong beds (A), firmly se cured to masonry, and between which the wheel revolves.

When used for pitching and trimming, the lock bolt connecting the two worm wheels is removed, and the pitch is given by the train of change wheels and division plate (A). The place of the slide rest is now taken by a headstock carrying two cutters, one for roughing, and the other for finishing.

The finishing rose-cutter is the counterpart of

the space between the teeth, and is transversed across, making both sides of the tooth alike.

The remainder of the arrangement will be obvious from the sketch. The same machine can be also readily arranged for cutting worm-wheel teeth, or for bevel gear.

The best form which can be given to the teeth of wheels is that which will cause them to be always, in regard to the power they mutually exert, in equally favorable situations, and, consequently, will give the machine the property of being moved uniformly by a power constantly equal. This would be accomplished by simple rolling contact, which corresponds with the case in which the teeth are infinitely small.

Definitions.

1. Spur gearing is that in which the pitch lines of the driving and driven wheel are in the same plane (fig. 90).

2. Bevel gearing is that in which the planes of the pitch lines of the driving and driven wheel