THEORY OF ELECTROMAGNETISM. 
707 
must be functions of 6 only. And further, by including these functions in — dljdd 
and — dl s /dd respectively—a proceeding that will not affect the equations of motion 
deduced from the form of L—we see that each of these quantities may be put equal 
to zero. With this extended meaning of L, then 
/= - a\/06> - SV 0 V\.(11), 
f s = - ax,/30 + [SU» 0 V\]« + *.(12). 
We now see also from eq. (10) that 
E = A + jljVcfe + \[ofds .(13), 
or 
E = jjjeds -f [j 'e 5 ds .(14), 
where 
e = A + 0f= X - 0d\/B9 - dSV 0 VA.(15), 
e t — K + @fs = K ~ Od\ s /d0 + 6 [SUz^ e VX] ff + 4 .... (16), 
so that e, e s may be called the intrinsic energy per unit volume and surface respee 
tively. 
Be. Frictional Forces, Conduction of Heat, and Dissipation of Energy. 
35. It has already [§ 27 (26)] been mentioned that a? is a function of 
6, ©; % V; K, H.(17). 
Of these ■'F and K are of the nature of velocities, and from the equation 4t rG = YVH 
the same may be said of H. Let us, then, briefly speak of them as “the velocities 
involved in x. Similarly in the general theory where x s is not assumed zero, it also 
will involve certain variables for like reasons called velocities. Let f £ s be the 
functions which are reciprocal (Routh’s £ El. Rig. Dyn.,’ 4th ed., § 410) wuth regard 
to © and the velocities, to the functions x and x s . Thus 
x + £ = - S© 0 Vx - SK k Vx - SH h Vx - <Jx£* .(18), 
* This seems a good opportunity to place on record a suggestion. There are some obvious objections 
to the method used, in the present and former papers of indicating the independent yariable of 
differentiation of a V or Q by an affix. It is somewhat hard to distinguish between C and c in the 
4x2 
