Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/151

Let $S_{1}$ be the first of these diaphragms, and let the line-integral of the force for a line drawn in the acyclic space from a point on the positive side of this surface to the contiguous point on the negative side be $\kappa _{1}$, then $\kappa _{1}$ is the first cyclic constant.

Let the other diaphragms, and their corresponding cyclic constants, be distinguished by suffixes from 1 to n, then, since the region is rendered acyclic by these diaphragms, we may apply to it the theorem in its original form.

We thus obtain for the complete expression for the first member of the equation

$\iiint \rho 'Vdx\,dy\,dz+\iint \sigma 'V\,dS+\iint \sigma _{1}'\kappa _{1}\,dS_{1}+\iint \sigma _{2}'\kappa _{2}\,dS_{2}+\mathrm {etc} .+\iint \sigma _{n}'\kappa _{n}\,dS_{n}$.

The addition of these terms to the expression of Green's Theorem, in the case of many-valued functions, was first shewn to be necessary by Helmholtz^[1], and was first applied to the theorem by Thomson^[2].

Physical Interpretation of Green’s Theorem.

The expressions $\sigma \,dS$ and $\rho \,dx\,dy\,dz$ denote the quantities of electricity existing on an element of the surface S and in an element of volume respectively. We may therefore write for either of these quantities the symbol e, denoting a quantity of electricity. We shall then express Green's Theorem as follows—

$M=\sum (Ve')=\sum (V'e)$;

where we have two systems of electrified bodies, e standing in succession for e₁, e₂, &c., any portions of the electrification of the first system, and V denoting the potential at any point due to all these portions, while e' stands in succession for e₁', e₂' , &c., portions of the second system, and V' denotes the potential at any point due to the second system.

Hence Ve' denotes the product of a quantity of electricity at a point belonging to the second system into the potential at that point due to the first system, and $\sum (Ve')$ denotes the sum of all such quantities, or in other words, $\sum (Ve')$ represents that part of the energy of the whole electrified system which is due to the action of the second system on the first.

In the same way $\sum (V'e)$ represents that part of the energy of

1. ‚Ueber Integrate der Hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen‘, Crelle, 1858. Translated by Tait in Phil. Mag., 1867.
2. ‚On Vortex Motion‘, Trans. R. S. Edin., xxv. part i. p. 241 (1868).