146 Mr. A. Gray on the Determination of the 
Let now the circuit of the coil be closed ; a retarding force 
due to the current induced in the wire, and, if the effect of 
self-induction be neglected, proportional to the angular velo- 
city, will act on the coil ; and the equation of motion for this 
case will be of the form 
S + *f-* ■••■.« 
For let I be the mean intensity of the magnetic field over 
the space occupied by the coil at time t, M the electromag- 
netic inertia (coefficient of self-induction) of the circuit for that 
position of the coil, R the total resistance in the circuit, /x 
the moment of inertia of the coil round a vertical axis passing 
through its centre, L the effective length of wire in the coil 
(that is, the length of wire in its two vertical sides), and b 
the mean half-breadth of the coil. If we call X the number 
of lines of force which pass through the coil at time t, and 
y the strength of the induced current in the coil at that in- 
stant, Ave have plainly 
N=&ILsin0-M y . 
The rate at which X increases per unit of time is therefore 
d^ ttt ndQ tt ■ /i^I d /lir > 
I =HL«^+H.-n* aF - a (M r ), 
and if 6 be small, and I be therefore supposed the same for 
every position of the coil, we have approximately 
ds 1TT cW a ,__ , 
-dt= hlL dt-jt (M ^ 
dN . 
But -=- is the electromotive force due to the inductive action ; 
at 
hence the current 7 is by Ohm's law given by the equation 
blLdO 1 d ,,, x 
It was assumed that the second term of this expression for 
7 would prove negligible in comparison with the first ; and 
this assumption was so far justified by the results of the expe- 
riments, which agreed fairly well with results obtained, for 
other instruments of the same pattern, by a modification of 
the second method described below. 
The couple due to the action of the field on the current 
is blLy ; and therefore, on the supposition of negligible self- 
induction, the retardation of the angular velocity of the coil 
at time t is 
d6 VI 2 L 2 d0 
dt ~ fill dt' 
