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2. The same problem, only the n'® being coupled with the first, 
so as to produce a closed chain of currents. 
For the sake of simplicity we will put equal the coefficients of 
selfinduction and the resistances for all circuits, the same assumption 
being made for the coefficients of mutual induction. 
1. Linear series of coupled circuits. 
Our case of an electromotive force existing in the first circuit 
and disappearing suddenly at the time ¢= 0, is analytically equi- 
valent to the case that at the time ¢—O the current is zero for 
E 
all circuits except for the first where its value amounts to 1, =—. 
: 
Putting the current in the first, second, third, ete. circuit 7,,7,,75..., 
ete.; and the coefficients of selfinduction JZ, the coefficients of mutual 
induction MW, the resistances 7, then we have the following set 
of simultaneous differential equations : 
Ae 4 = —0| 
iel er 
; L di, Vv di, u di, 0 
ETE 1 oe 
rat 4 
lath ar Ret: ay ar Bay tare 
din—1 din di 
bart EL M — M — =0 
it dt 
din din—1 
Ie : = M — 0 
veen dt id dt J 
the initial condition being 
for t= 0 ET == Sie ee 
ae 
In order to obtain the solution we take 
== oe 
t= ae 
dy = Oy, EP! 
Further, introducing 
r+ pL 
Pp 
and substituting these expressions in the differential equations, we 
obtain the homogeneous equations 
