certain Bessel Integrals. 341 
and from (25) 
M=200<w(l0- \ + \, + ± i+ Jr + + i - | A. + . . .). 
Using the terms shown, we find that this gives 
M/wW = 19057-28 (26) 
Further, by calculating a few more terms, we easily see 
that the result in (26) is correct as far as the figures shown. 
Other series for this case are those of Maxwell * and 
Heaviside f, while a complete expression in elliptic functions 
has been given by Cohen }. In the last case, although an 
exact theoretical expression is found, yet in practice the 
accuracy depends upon tables of elliptic functions and upon 
the result of long and complicated calculations. These three 
expressions have been compared numerically by Rosa and 
Cohen § for the case used above ; thev give the following 
results for M/tA 2 :— 
Maxwell's series 19057-25 
Heaviside's series 19067*08 
Coheir's elliptic-function formula... 19057*36 
Comparing these with the result given in (26) we infer 
that the two latter formulae do not give better results than 
Maxwell's, at least when the ratio of length to diameter is 
large ; Cohen's formula is applicable to all values of this 
ratio, but it is not suitable for calculation. The series given 
in (25) appears somewhat simpler than Maxwell's ; it is con- 
vergent for all coils with the length greater than the radius 
of the outer coil, and as one knows the general term of the 
series the result can be calculated to any required degree of 
accuracy ; it can easily be verified that the series converges 
quite rapidly even for coils whose length is not much greater 
than their breadth. 
* Maxwell, ' Electricity and Magnetism,' vol. ii. § 678. 
t Heaviside, ' Electrical Papers,' vol. ii. p. 277. 
X Cohen, Bulletin of the Bureau of Standards, vol. iii. p. 295 
(1907). 
§ Rosa and Cohen, Bulletin of the Bureau of Standards, vol. iii. p. 316 
(1907). 
