In order to find the pressures arising from the weights r and w, the ;um of those weights must be considered as applied at a point a in the axis of the machine, where that axis would be cut by a vertical plane passing through the common centre of gravity of the two a-eights : let c and c' be points in which the axis is cut by vertical planes passing through the respective centres of gravity of r and w ; ',hen, in order to find a, we have, by mechanics, purposes. The engines used for this purpose vary in their construc tion according to the wants or caprice of the artists who use them. We shall content ourselves with giving a description of the engine commonly employed, with a few remarks on the kind of tools used for cutting the spaces between the teeth, which operation is usually termed cutting the teeth of a wheel, although in reality the teeth are those portions of the metal which are left standing. We shall, however, employ the common phrase, as it will perhaps be best understood by all who feel an interest in the art.
Description of the engine commonly used :—AAAA a strong frame If the wheel and cylinder are in a state of motion about their mathe matical axis, the pressure on the supports will evidently be diminished by the force with which the common centre of gravity of the weights P and w tends to descend ; the valne of this force is investigated in treatises on dynamics.
If two wheels and cylinders are connected together by a string b ad f (in fig. 1), or by teeth in the circumferences, as in most forms of rack-work, the ratio between the power P and the resistance w, in the case of equilibrium, may be determined by the same rule as would be employed if those weights were at the opposite extremities of a double lever of the first or second kind. For the power P may be conceived to be applied at A perpendicularly to the semidiameter ea, and it will be in equilibrio with a resistance at a, perpendicular to CA ca, which may be expressed by P. : let this be represented by p.
Now this force at a may, in consequence of the string passing round the axle cn and the circumference of the wheel Bs, be conceived to he a moving power applied at b perpendicularly to EA ; and this will be in equilibrio with a resistance w at F, acting perpendicularly to FE, which EL may be expressed by p — therefore, substituting in it the above