1921-22.] The Faraday-Tube Theory of Electro-Magnetism. 235 
, the .r-cornponent of electric displacement, will be constant at a point 
on the tube, or 
\ 8 t 8 xJ 
Also, if dy is the ^-component, we shall have 
dy dij 
dv 
And thus 
7 dy . 
dy — X 
OX 
d“ — dy^^ + dy^ 
= dg^-y 1 + 
\dx) J 
(43) 
The momentum per unit length along the tube is 
Vd,B = fxVdj^Vqd 
== a(q . d -d . qd^) . 
d 
Multiply this by ^ to find the value appropriate to unit length along 
the ,^-axis, and, taking the ^-component, we have 
f f I + J£)d - dSjt V! udM ^,+ I 
"dx r/\ 
m 
dxj dx\ 
' - (I)’} 
= ydgg^ 
(44) 
Hence the rate of change of momentum in the ^-direction per unit 
length along the ^r-axis is 
^ )ydg^d. = fxd 
dt dx 
dt 
dh) . dhf ' 
_ dt^^ dxdt_ 
(45) 
The force to be equated to this arises from the quasi-tension 
E + VqB = ^ + /iVqVqD 
of which the i/-coinponent is 
AY! 
K ^dx ' ^\dt. ^dxJ I 
\dt dxJ dxi 
"^dx I 
dt dx) J 
1 7 
-f^dgg -- + /X 
iv OX 
di/ dy 
-A + u~^ 
dt dx 
'tdx - y-dg^id 
ox. 
_^jdy J dy 
'K + ^ "“07 
( 46 ) 
