THEORY OE ELECTROMAGNETISM. 
72:3 
The second thing to notice is that the E of equation (29) is not what is usually 
known as the electromotive force. The physical fact that is usually stated by saying 
that E = ItK, must with the present notation be stated by saying that E = 0, since 
— IlK appears on the right of equation (29) as a part of the term — c Vx. This, of 
course, is due to the fact that E of equation (29) is physically defined as the part of 
the electromotive force not depending on friction. 
D. Change of Variables in l, X, and x. 
51. In what follows with reference to change of variables, we shall always speak as 
if the change had reference only to l. Exactly similar reasoning applies to similar 
changes of variables in any other function such as X, x, or a part only of any one of 
these. There is, indeed, no reason why the function should be a scalar. 
So far, l has been assumed an explicit function of the list of variables (25), § 27. 
These are by far the most convenient variables for most mathematical operations, and 
we shall continue as often as otherwise so to regard l. For many physical interpreta¬ 
tions, however, it is necessary to regard l, or a part of it, expressed in terms of other 
variables. Consider, for instance, air as a dielectric. This will be taken account of by 
supposing l to contain a term quadratic in d. Suppose, now, we compress the air till 
its density is (say) doubled. We know as a matter of experimental fact, that the 
specific inductive capacity will not thereby be largely altered. This will mean, not 
that the quadratic expression in d is but slightly altered in form, but that the equal 
expression in d' is thus slightly altered. Moreover, to express simply the fact of 
electric and magnetic isotropy of fluids requires that the independent variables should 
be the dashed letters. Let then 
where 
Ids = I'ds = l"ds or l = mV = ml" . (l), 
l is an explicit function of 0, © ; p ', p , T'; d, D, C, H 
v » „ 0 , e # ; p\'p' t \ k; d', D', C', H'; q }. . ( 2 ). 
I" » „ 0, 0 " ; p, p, T ; d", D", C", H" 
Defining X', X" similarly, it may be said here what will appear incidentally later, 
that X' and l', and again, X" and l", are related to one another exactly as are X and l ; 
i.e. [§ 46, eq. (11)] 
V + X' = - DV 3 - SC' C V' V - Sir H V' V 1 
V’ + X" = - DV 3 - SC" c V"*" - SH" h V'T'J ’ 
where C V' is put for C ,V, &c. 
4 z 2 
