223 
PROCEEDINGS OF SECTION A. 
As an example consider the series 
/ M = * - f + f - 
absolutely convergent for — 1 < x < 1 
+ 
we shall have- 
f l (x) = 1 — X 2 ■+■ # 4 — # 6 + 
1 
1 + X 2 
therefore 
arc tan. x 
( 7 r 7j- \ %XP 
taking the arc between — — and + - ■ 1 and the series 2 ( — l) n — 
It 2x J 71 
represent are tan. x for — 1 < x < 1. 
x n 
With regard to x = 1 the series 2 ( — 1)* — is still convergent 
(only), but the notions we have given are no longer sufficient to study 
it ; we must have recourse to Abel’s theorem (complete). This theorem 
7 r 
shows that the series still represents arc tan. x for x = 1 , i.e., — . 
“ad 
It would be easy to extend the preceding results to series whose 
terms are complex quantities. 
Any series 
a Q + a x x + a 2 x 2 + .*.• + a n x n + 
absolutely convergent for all points inside a circle of radius a defines 
in that circle (i.e., for all the points inside that circle) a continuous 
function of x capable of derivation and integration. 
This is Briot and Bouquet’s theorem. (Recherches sur les series 
ordonnees suivant les puissances croissantes d’une variable imaginaire 
— C.B. 1853, p. 26 i and p. 331). 
6.— THE ENERGY OF THE ELECTRO-MAGNETIC FIELD. 
By PROFESSOR BRAGG, M.A . 
The expressions for the energy of an electro-magnetic field are 
deduced from certain principles laid down by Maxwell. In the 
deduction it is not at all necessary to employ, as is often done, the 
idea of the “magnetic shell.” This idea is not only highly artificial 
but is apt to lead to wrong impressions. The shell cannot be made of 
a number of small and similar steel magnets placed side by side. It 
consists of two layers of magnetism, separated by a thin uniform layer 
of air. In fact it is, as I pointed out in an address to this Section at 
Hobart, the magnetic analogue of the electrostatic plate condenser. 
In order to understand how the expressions can be obtained 
without using the magnetic shell, it is, I think, a help to use first an 
analogy. 
