Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/191

the investigation depends on purely mathematical considerations depending on the properties of lines in space. The reasoning therefore may be presented in a much more condensed and appropriate form by the use of the ideas and language of the mathematical method specially adapted to the expression of such geometrical relations—the Quaternions of Hamilton.

This has been done by Professor Tait in the Quarterly Mathematical Journal, 1866, and in his treatise on Quaternions, § 399, for Ampère's original investigation, and the student can easily adapt the same method to the somewhat more general investigation given here.

523.] Hitherto we have made no assumption with respect to the quantities $A$, $B$, $C$, except that they are functions of $r$, the distance between the elements. We have next to ascertain the form of these functions, and for this purpose we make use of Ampère's fourth case of equilibrium. Art. 508, in which it is shewn that if all the linear dimensions and distances of a system of two circuits be altered in the same proportion, the currents remaining the same, the force between the two circuits will remain the same.

Now the force between the circuits for unit currents is ${\frac {dM}{dx}}$, and since this is independent of the dimensions of the system, it must be a numerical quantity. Hence $M$ itself, the coefficient of the mutual potential of the circuits, must be a quantity of the dimensions of a line. It follows, from equation (31), that $\rho$ must be the reciprocal of a line, and therefore by (24), $B-C$ must be the inverse square of a line. But since $B$ and $C$ are both functions of $r$, $B-C$ must be the inverse square of r or some numerical multiple of it.

524.] The multiple we adopt depends on our system of measurement. If we adopt the electromagnetic system, so called because it agrees with the system already established for magnetic measurements, the value of $M$ ought to coincide with that of the potential of two magnetic shells of strength unity whose boundaries are the two circuits respectively. The value of $M$ in that case is, by Art. 423,

the integration being performed round both circuits in the positive direction. Adopting this as the numerical value of $M$, and comparing with (31), we find