THEORY OF ELECTROMAGNETISM. 
705 
Equation (1) can be put in a more convenient form for our purpose. We have 
0L /0L d 3L\ ~ ^ /3L ~ , 3L » \ 
8L = It 2 aj s 2 + 2 f - It fi s * + 2 (a* ^ **■) 
With the restrictions just mentioned, we have 3L/0</> = 0 and S<3? = 0. Elence 
s r d x dL s >v/ 0L d3L )s 
1 
n d _ /0L d 0L\ - , 
~~ ^7t X ( Sqa ’ Sqb ’ * ' •) + -Jtdq) Sq J 
y • • • 
(*)> 
where 2.T (S q a , Sq/j . . .) is to be defined as the function which appears under the 
operator d/dt when SL is expressed as the sum of two quantities, one of which is a 
linear function of the variations of the retained coordinates, and the other is the rate 
of variation of a similar function. We now have 
A = 2£ (q a , q h . ) — L 
. . (3). 
We shall show how, for our particular system, % (q a . . .) can be expressed in the 
form 
X (?«, q_h • • •) = f 11 t ds + 1 1 t ds . (4), 
where t, t s are functions of the same independent variables as l, l s . 
It is convenient to call X (q a . . .) the whole modified kinetic energy, and t, t s the 
modified kinetic energies per unit volume and surface respectively. And, similarly 
putting 
X = 2t-l, K = 2p - l s .(5), 
we shall call A and \ s the free energy* per unit volume and surface respectively. We 
thus have 
A = jfjxcZs + [jx/7s.(6). 
We shall then assume that the energy in any finite region is the integral on the 
right of this equation for that region. The surface integral in this case, of course, 
only applies to surfaces of discontinuity (as to physical quantities) in (his region, and 
not to the true boundary of the region. 
* This term is adopted as a translation of Helmholtz’s ‘/me Energie’ (‘ Wiss. Abh.,’ II., 959). It is 
not, of course, the same as the intrinsic energy ■which we are about to determine by a method analogous 
to Helmholtz's. 
MDCCCXCII.— A. 4 X 
