Page:On Faraday's Lines of Force.pdf/55

Rh With respect to the history of the present theory, I may state that the recognition of certain mathematical functions as expressing the “ electro-tonic state of Faraday, and the use of them in determining electro-dynamic potentials and electro-motive forces is, as far as I am aware, original; but the distinct conception of the possibility of the mathematical expressions arose in my mind from the perusal of Prof. W. Thomson’s papers “ On a Mechanical Representation of Electric, Magnetic and Galvanic Forces,” Cambridge and Dublin Mathematical Journal, January, 1847, and his “ Mathematical Theory of Magnetism,” Philosophical Transactions, Part I. 1851, Art. 78, &c. As an instance of the help which may be derived from other physical investigations, I may state that after I had investigated the Theorems of this paper Professor Stokes pointed out to me the use which he had made of similar expressions in his “Dynamical Theory of Diffraction," Section 1, Cambridge Transactions, Vol. IX. Part 1. Whether the theory of these functions, considered with reference to electricity, may lead to new mathematical ideas to be employed in physical research. remains to be seen. I propose in the rest of this paper to discuss a few electrical and magnetic problems with reference to spheres. These are intended merely as concrete examples of the methods of which the theory has been given; I reserve the detailed investigation of cases chosen with special reference to experiment till I have the means of testing their results.

EXAMPLES.

I. Theory of Electrical Images.

The method of Electrical Images, due to Prof. W. Thomson, by which the theory of spherical conductors has been reduced to great geometrical simplicity, becomes even more simple when we see its connexion with the methods of this paper. We have seen that the pressure at any point in a uniform medium, due to a spherical shell (radius=$$a$$) giving out ﬂuid at the rate of $$4\pi Pa^2$$ units in unit of time, is $$kP\frac{a^2}{r}$$ outside the shell, and $$kPa$$ inside it, where r is the distance of the point from the centre of the shell.