COFFIN. — THE SELF-INDUCTANCE OF COILS. 793 



But the mean square value of an harmonic function is one half the 

 square of its amplitude, and in working with imaginary expressions 

 the square of the amplitude can always be obtained by multiplying 

 any expression by its conjugate. Hence 



T 



P = - Cpdt = ^lT (13) 



o 



where / denotes the conjugate to /. 

 Therefore 







In general, 



F =lfff H ' d "- < 15 > 



00 



if [a, the magnetic permeability, is assumed equal to 1 everywhere, which 

 becomes, in this case, 



and 



per unit length of coil 



• j H 2 27rrdr 

 o 



j 



= T x + T, 



2 rdr 



(16) 



ia>t 



where, in T 1} H= JI e 



and in T 2 , H= eq. (8). 



We may write 



- Crdt = ^ frdr- f H*dt = l ClIHrdr. 



*o o 



For T x this becomes f -|^ j . (17) 



To obtain the integral T 2 first put in equation (8) h (1 + i) for ih. 

 Then, calling h (r a — r) = y, 



