230 
DR. J. HOPKINSON AND MR. E. WILSON ON 
represented as an armature reaction which vanishes at the moment when the current 
in the armature vanishes. 
To state the matter in the form of an equation, let E be the electromotive force of 
the machine on open circuit, R the resistance of the armature circuit, x the current 
in the armature, T the periodic time, then it is assumed that 
Rx = E — (Lx )’; 
E being independent of x, L being, if you please, a coefficient of self-induction constant 
or variable, or, if you prefer it, Lx representing the change in the induction through 
the armature due to the current in the armature, vanishing with x. 
It is easy to see that this statement is true in some cases. For example, it is very 
nearly true in the older machines with permanent magnets. Or imagine a machine 
without iron in the magnets or armature, consisting merely of two circuits—one the 
magnet circuit, the other the armature circuit—movable in relation to each other. 
If the current in the magnet circuit is kept precisely constant, either by inserting a 
great self-induction in its circuit external to the machine, or by inserting such a 
resistance and using so high an electromotive force that any disturbing electromotive 
forces are inappreciable compared with it, the preceding statement is strictly accurate. 
But if the magnet current is not forced to be constant the problem is more 
complicated. 
Stating the matter in the language of self- and mutual-induction, let x and y be 
the currents in armature and magnet circuits, II and r their resistances, L and N 
their self-induction, M sin 2nt/T their mutual-induction, E the constant electromotive 
force applied to the magnet, the equations for the system are :— 
Rx = — M (y sin 27rt/T) < — Lx’ 
ry = F — M (x sin 2nt/T)’ — N y [ 
T1 
Tiese equations can be solved by approximations if the variations in the value of 
y are small. 
First, 
F MF 2tt E cos 27rt/T + 2ttL/T sin 2 tt//T 
y = — ; x — — 
rn 
V 1 
E- + (2ttL/T) 2 
Second, substituting this value of x, we obtain 
ry = F + 
M 2 F 
:7T 
/ T 
E 2 + (2ttL/T) 
E . 47T t 7T L / 47tA 1 • , T 
n2 , — sill — -F Y \ l ~ cos ~¥) \ ~ 
Tin's gives periodic terms in y, the period being one-half the period of the mutual 
induction. 
Introducing these terms into the first equation, we see that the term in 27 rt/T in 
