728 
MR. A. McAULAY ON THE MATHEMATICAL 
Note that l 0 ' does not stand towards l 0 in the same way as V towards l, as appears by 
the equations 
ml' = l, Iq = l 0 .(28). 
In utilising equation (21), then, the analogue of V will be TIius the part 
contributed by l' Q to the terms [<£'] — V on the right of equation (21) will be 
— 2yC = — 2m -1 x®oVH“ m ~ 1 h'x li P~ 1 X ~~' m ~\> [equation (10)], 
= -2m- 1 x ffl 0 , x', 
since [former paper, equation (39)] 'Sf -1 = X -1 X -1 - Assuming, then, <£ 0 to be of the 
first class of § 9, and defined by 
^=-2 ai,; .( 29 ), 
equation (21) gives 
2 4> — 2</>' 0 + (/r 0 H'Y47r — 47 tK 0 _1 d' 3 ) 
-zs(r'yr+ a'yr) + zv {r\ ) t vt - o -'( )Xn 
+ { X V^(/- 1 ( ) q + qVx ( )v- ( l~ 1 } .(-80), 
where l\ cr', t, and r) have exactly the same meanings as before, so that, indeed, 
r = l'olm+ 2 7 rK 0 - 1 d ,:J - / x 0 H /3 /87r.(31). 
E. Connection hetioeen E and e. 
56. So far it has been assumed that there are two independent kinds of external 
force denoted by E and e, and by E 4 and e*. This is contrary to the usual custom, but 
seems to me to be a necessary consequence of assumptions always made as to the 
difference in nature between what is ordinarily called the displacement current and 
the conduction current. 
The independent variables required to fix the electric state at a point have for 
mathematical convenience been taken as D and d. These are, perhaps, not the most 
natural. It would seem from the ordinary views as to the two kinds of current as if 
the dielectric displacement d, and the conduction displacement k are the most natural. 
Moreover, I believe it is generally held that d has exclusively to do with the potential 
energy of electrification. It seems, then, likely to lead to correct results to assume 
that if d and k were taken as the independent coordinates, there would never be any 
external force of type d. 
As this conclusion may seem open to question let us put the matter in a different 
