Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/266

 Rh 615.] These may be regarded as the principal relations among the quantities we have been considering. They may be combined so as to eliminate some of these quantities, but our object at present is not to obtain compactness in the mathematical formulae, but to express every relation of which we have any knowledge. To eliminate a quantity which expresses a useful idea would be rather a loss than a gain in this stage of our enquiry.

There is one result, however, which we may obtain by combining equations (A) and (E), and which is of very great importance.

If we suppose that no magnets exist in the field except in the form of electric circuits, the distinction which we have hitherto maintained between the magnetic force and the magnetic induction vanishes, because it is only in magnetized matter that these quantities differ from each other.

According to Ampère's hypothesis, which will be explained in Art. 833, the properties of what we call magnetized matter are due to molecular electric circuits, so that it is only when we regard the substance in large masses that our theory of magnetization is applicable, and if our mathematical methods are supposed capable of taking account of what goes on within the individual molecules, they will discover nothing but electric circuits, and we shall find the magnetic force and the magnetic induction everywhere identical. In order, however, to be able to make use of the electrostatic or of the electromagnetic system of measurement at pleasure we shall retain the coefficient μ, remembering that its value is unity in the electromagnetic system.

616.] The components of the magnetic induction are by equations (A), Art. 591, {{numb form | $$ \left. \begin{align} a = \frac{dH}{dy} - \frac{dG}{dz}, \\ b = \frac{dF}{dz} - \frac{dG}{dz}, \\ c = \frac{dG}{dx} - \frac{dG}{dy}. \end{align} \right\} $$| }} The components of the electric current are by equations (E), Art. 607, {{numb form | $$ \left. \begin{align} 4\pi u = \frac{d\gamma}{dy} - \frac{d\beta}{dz}, \\ 4\pi v = \frac{d\alpha}{dz} - \frac{d\gamma}{dx}, \\ 4\pi z = \frac{d\beta}{dx} - \frac{d\alpha}{dy}. \end{align} \right\} $$| }}