340 Dr. T. H. Havelock on 
n 2 be the number of turns o£ wire per unit length on the two 
cylinders. Then if we write x—x'\ for the absolute value 
of x—x\ we have from (21) 
M 
J 00 rh f*h 
JifXa^XfydX] dx\ dx'e-^ x ~ x ' 1 
o J—h J-h 
= 167T 2 aK^J o <[£ - ~ + ^ e~ 2hK | Jj(Xa) J&Qdk. 
With b > a and h > b, we use the series (3) . (4) and (12) ; 
thus we find for the coefficient of mutual induction of the 
two coils : 
M-8-«^.[3-| + A(f-)" +a J g (f)V 1 fi B (f)V • • • 
This gives an expression for M which is easy of calculation 
and rapidly convergent ; moreover, from (4) and (12), 
additional terms in the two series within the brackets in (25) 
can be calculated if required from the general terms 
U.3.5....(2r-3)P(2r-l) / a_f 
2 a *+ 1 r!(r + l)l \bJ 
and 
We consider as a numerical illustration a case which has 
been used in comparing other similar series, namely : — 
a = 5 cm. ; b = 10 cm. ; 7i = 100 cm. ; n 1 =n 2 =n. 
Then we have 
^ _ 10 ^ ^ a _1 
b~ ; h~ib ; b~r 
