702 
MR. A. McAULAT ON THE MATHEMATICAL 
Nothing is here said about l s and x s , as it has been thought advisable to see how 
much can be explained without their aid. When we come to consider electrostatic 
contact potential difference—which for brevity we will in the future call contact-force 
—it will be found necessary to suppose l s not zero. For ordinary friction, also, x s 
must not be zero. The above assumptions will enable us to take account of (1) all 
Maxwell’s results, or results corresponding thereto; (2) the stresses, &c., resulting 
from variation of specific inductive capacity and magnetic permeability with strain 
and temperature ; (3) thermoelectric, thermomagnetic, and the Hall effects ; (4) many 
purely mechanical results whose details will be reserved for future treatment. They 
do not enable us to take account of (l) sliding friction ; (2) electrolysis ; (3) hysteresis 
and similar phenomena ; (4) contact-force. All these, however, except (3), can be 
taken account of by slight additions to our present assumptions, as in the case of (4) 
will appear later. 
The object of limiting as above the number of the independent variables entering 
into l and x is to free the mind from unnecessary vagueness. Moreover, the above 
assumptions are in one sense necessarily simpler than those made by Professor J. J. 
Thomson ( ‘ Applications of Dynamics to Physics and Chemistry,’ 1st ed., chap, vii.) to 
explain thermoelectric and thermomagnetic effects, in that the only quantity whose 
space-variations appear above in l or x is 6, a statement not true of Professor 
Thomson’s assumptions. With regard to the forms of l and x as functions of their 
independent variables, it is simplest at present to make no restrictions. 
28. I am a little doubtful whether writers on the present subject recognize two 
semi-independent electric displacements at every point, but, as already remarked, it 
seems to me to follow, as a matter of course, from the assumption of two independent 
currents. The independent variables which have [§ 27 (25)] above been chosen to 
take account of these are d and D, though, of course, d and k or k and D might have 
been chosen instead. Maxwell generally, but not quite without exception, seems to 
use the symbol D for what I have called d. I thought, however, that I should be 
following the usual custom of subsequent writers by using D for the whole displace¬ 
ment. 
If there be two independent electrical displacements, it would seem as though we 
must assume, at any rate provisionally, the existence of two independent external 
electromotive forces of type D and d. These we shall denote by — E and —■ e respec¬ 
tively. This, of course, means that the work done by the said external forces 
at the element ch, while D, d change to D-f c/B and d + c/d respectively, is 
(SE c/D -f- Se c/d) ch. We shall also assume external surface forces of these types 
— — e.,, external ordinary forces E and F, y per unit volume and surface of the 
present position of matter, # and an external stress <4>, <3> being a self-conjugate linear 
vector function of Class I of § 9 above. This last statement means that the real 
' In the former paper y meant the force per unit volume of the standard position of matter. The 
change has been made, since the equations of this paper are thereby simplified. 
