APPENDIX.

THE important theorem of reciprocity, established in Art. 6, may be put in a clearer light by the following demonstration, which is due to Professor Maxwell.

Let A, B be any two points on a closed conducting surface, and let a unit of positive electricity be placed at a point Q, within the surface, then a unit of negative electricity will be so distributed over the surface that there will be no electrical force outside the surface, and the potential outside it will be everywhere zero. The potential at any point P within the surface, due to the electricity on the surface, is a function of the positions of P, Q, and of the form of the surface.

Denoting this by G, it is required to shew that G = G'}, or that the potential at P, due to the distribution on the surface caused by a unit of positive electricity at Q, is equal to the potential at Q, due to the distribution on the surface caused by a unit of positive electricity at P.

Let X be any point outside the surface. The potential there is zero, hence

ᎵᎪ

A

where p is the density and dS, the element of surface, at any point A of the surface, and the integration is extended over the whole surface.

Now if we consider a unit of positive electricity placed at P, and if ρ be the density on an element dS at B, we shall have, similarly,

PB

for all points outside the surface, or on it, since the potential is zero on the surface.

Let X be on the surface, say at A, this equation becomes

1

1

Σ dSB PB BA PA
+ = 0.

Hence, substituting in equation (2) we get

and as this is the same as we shall obtain for G, the property is proved.

Note to Art. 10, pp. 50, 51.

a

The equation 4 (r) == ↓ (~) proved on p. 51, may be ex

pressed in words as follows. Let O be the centre of a sphere of radius a, and A, B, two points each of which is the electrical image of the other with respect to the sphere (i.e. let O, A, B be in the same straight line, and OA.OB=a2), then, if electricity be distributed in any manner over the surface of the sphere, the potential at A is to the potential at B as a is to OA or as OB is to a.

For, if a point P move in such a manner that the ratio BP to AP is constant (= λ suppose) it will describe a sphere, and if C, C' be the points in which this sphere cuts AB,

And potential at A: potential at B:: :

1
PA PB

1

: λ: 1.

Hence the theorem is proved.

The laws of the distribution of electricity on spherical conductors have been geometrically investigated by Sir William Thomson in a series of papers published in the Cambridge and Dublin Mathematical Journal. See also Thomson and Tait's Natural Philosophy, Arts. 474, 510.

Note to Art. 12, p. 68.

In the case of a straight line uniformly covered with electricity, the form of the equipotential surface, and the law of distribution of the electricity over the surface may be investigated as follows.

Denoting the extremities of the straight line by S, H, we know that the attraction of the line on p' may be replaced by that of a circular arc of which p' is the centre, and which touches SH, and has Sp', Hp' as its bounding radii. Hence the direction of the resultant attraction bisects the angle Sp'H, and the equipotential surface is a prolate spheroid of which S, H are the foci.

Again,

dv

dw'

is the resultant force exerted by the straight line,

or by the circular arc, and therefore

Sp'. Hp' sin Sp' H