338 
Dr. T. H. Haveloek 
on 
§ 2. Mutual Induction of Single-layer Coil and 
Coaxial Circle. 
^ If we have two coaxial circles of radii a and b, with a 
distance c between their planes, we have their coefficient of 
<- c 
M 
mutual induction 
^ ' - o Jo 
iven by 
,,ir ab cos ((f) 
</>')##' 
= \ j a&cos((/>-<£0##'| 
Vc 2 + a 2 4- b*-2ab cos (<£-<£') 
J (X \A** + &* - 2a6 cos (</> - <£') ) ^ 
-Ac. 
= 47r 2 a& 
J 1 oo 

^(Aa)^^/))^ (21) 
From the integrals (10) and (19) in the previous section 
series for M could be obtained suitable for c large or small 
compared with a or b. Further, if we have a single-layer 
solenoid of length 2h, radius b, and n turns of wire per unit 
length together with a concentric coaxial circle of radius a, 
less than b, we obtain their coefficient of mutual induction 
by integrating (21) with respect to c between the limits —h 
and + h. Hence we obtain 
M = Sir^ab 
n\ (l — e~ 
Jo 
M )Ji(Xa)J!(X5)^. 
(22) 
Using then the integrals given in (11) and (20) we have 
series suitable both for long and for short coils. However, in 
the latter case the difference between the radii of the coil 
and the circle must be small compared with one of them, and 
unless this holds series already in use probably give a better 
approximation than those obtained from (22) *. Finally, if 
* Cf. E. B. Rosa, Bulletin of the Bureau of Standards, vol. iii. p. 209, 
1907. 
