THEORY OF ELECTROMAGNETISM. 
703 
stress-function is not <b, but <t>', i.e., that the force exerted on a region at the element 
of its boundary is clS = d%'. 
29. The meaning of “external” may be defined as “not included in the form of 
L.” Thus, the external forces include (1) all frictional forces given by X ; (2) forces 
that, though not now included in the form of L, can be so included by generalising 
the meaning of l and l s , so as to explain electrolysis, contact-force, and chemical 
phenomena; (3) forces that, through present ignorance we cannot include in X or L, 
though they should be so included. Thus, for.instance, the external stress <£> may be 
supposed to be due entirely to viscosity and elastic fatigue, and the first of these will 
be accounted for by X. 
30. SD' is due partly to variation of strain and partly to SD ; let S'D' be the latter 
part. E and e are assumed to be intensities. Hence (§ 8, Prop. II.) 
(SE SI) + Se Sd) ds = (SE 7 SD 7 + Se' Sd 7 ) dd . . . . (27). 
A similar theorem is supposed to hold with regard to E,, e,, viz.: 
(SE,. SD + Se, Sd) ds = (SE', S'D 7 + Se', S'd') ds . . . (28), 
from which, since [§ 7, eq. (6)] S'D' = m~ l x SD, and 
ds Ids = T dt'J T dt = mT x '- l \Jv = mT“ yiL/ 
E'^x'-'E/Tx" 1 ^, E, = x'BVTx'U^.(29), 
and similarly for e„ e', s . 
31. We must distinguish carefully between the independent variables of an element 
of matter which are given in the two lists (25) and (26) of § 27 and the independent 
variables of the system in general. These last consist only of 
0, p, d, D .(30), 
for every element of matter, for when these last and their time-rates of variation are 
assigned for all space, all the other quantities are determined. [It is not quite correct 
to talk of D as an independent variable on account of the equations of condition (7) of 
§ 24 -J 
To enable us to develop the consequences of these fundamental assumptions, a 
digression on dynamics and thermodynamics must be made. 
