THEORY OF ELECTROMAGNETISM. 
733 
Next note that by d/dt, or a dot, is denoted differentiation with regard to time, 
which follows the motion of matter, and by d/dt, a differentiation at a fixed point of 
space, so that djdt is commutative with V, but not with V', and d/dt with V', but not 
with V. Hence, as is well known, 
d/dt =-SpV. + d/dt . ( 21 ). 
Now, by equation (20), 
E ' 0 - x'-'Eo = - X - WA ’)/dt - V'v 
= ~ A' - x'-'x'A' - V>. 
Now [former paper, equation (25)], 
x'co = — Vfiup'y, 
so that 
“ X ,_1 X' A ' = x ,_1 V 1 SA'p , 1 = V'jSA 'p\, 
and, by equation (21.), 
- A' = - dA'/dt + SpV . A', 
therefore 
E' 0 = - dA'/dt, + SpV . A' + V'iSA'p 7 ! - V'v 
= - dA'/dt + SpV . A' - V'-lSAV - V' (v - SA'p') 
= - dA'/dt + Yp'WA' - Vz 
— - dA'/dt + Yp'B - Vz, 
where 
2 = v — SA'p' .(22). 
This proves equation (6). Of course, this more complicated form of equation (20) 
is necessary for some purposes, but the simpler form is more useful in discussing the 
general theory. From the simpler form, indeed, we may see at once that Maxwell’s 
result must follow, since it implies the truth of the principle from which he deduces 
his result. That principle is (‘ Elect, and Mag.,’ 2nd ed., § 598) that the line integral 
of E 0 round any closed curve moving with matter equals the rate of decrease of the 
line integral of A' round the same curve. Since both E 0 and A are intensities, this 
may in our notation be expressed by saying that the line integral of E 0 round the 
corresponding fixed curve equals the rate of decrease of the line integral of A round 
the fixed curve. This last is clearly insured by the equation E 0 = — A — Vv. 
Thus in the results contained in equations (5) to (11) the present theory is in 
complete agreement with Maxwell’s. Equations (14), (15), (17) may be taken as 
