Page:Scientific Memoirs, Vol. 1 (1837).djvu/520

at the same distance, and that the branches themselves be exactly cylindrical. Filing them into shape will perhaps have the disadvantage of hardening too much the surface of the iron, and of rendering it less apt to receive and to part with the magnetism. The form proposed offers a further inconvenience, in the application of the copper wire helices, which have to be previously bent on another cylinder of the same dimension. These helices ought very nearly to touch the bars, which should be covered with silk on account of the insulation which is necessary. In future an arrangement similar to the one in fig. 2 will be preferred, in which $f$ are the fixed bars, and $m$ the bars moveable around the axis $a$. We shall have the advantage of being able to employ cylindrical bars of soft iron, such as may be had of all dimensions in the shops. It will only be necessary to cut them into equal pieces, and the helices may be strongly wound round the bars by means of the lathe.

5.

As the magnetic attraction decreases rapidly as the distance increases, the integral $\int _{0}^{a}M\,ds$ will always be such a function of the amplitude $a$, that its value will not greatly differ from a constant, $a$ being rather considerable. Admitting, for an instant, that the magnetic attraction is in an inverse ratio to the squares of the distances, we shall have $\int _{0}^{a}M\,ds=\int _{0}^{a}{\frac {f\,ds}{d^{2}+s^{2}}}={\frac {f}{d}}\;{\mbox{arc}}\,tg{\frac {a}{d}}$, $d$ being the distance of the magnetic centres when the bars are placed the nearest possible; thus $d$ being very small with respect to $a$, $\int _{0}^{a}M\,ds={\frac {f\pi }{4d}}\;{\mbox{or}}\;={\frac {nf\pi }{4d}}$, representing the number of bars. We shall then have for the action of the motor, during one entire revolution, the expression ${\frac {n^{2}f\pi }{4fd}}$. The radius of the circle upon which the bars are arranged does not enter into this expression; and for a stronger reason it will not enter into any of the other expressions, if the attraction still decreases more rapidly than the inverse ratio to the square of the distance. Thus the size of the circle for the same number of bars scarcely adds anything to the action of the motor.

I conceived that the system of bars, which in my apparatus are fixed, might also be rendered moveable. The rotation of the two systems will then be in a contrary direction and have the same velocity, the masses being equal. These two motions might be combined by means of conical wheels, in order to produce the motion of a second axis of rotation intended for the work. The action of the motor, during the amplitude $a$, that is to say from one meeting of the poles to the other, would be as above ${\frac {nf\pi }{4d}}$, but the poles meeting each other $2n$ times in one revolu-