IN CYLINDRICAL CONDUCTORS, ETC. 
85 
When s is even these integrals are zero, and when s is odd have the values 
4 
s+ 1 
1 + 1 
+ k + 
+ \ 
Neglecting terms beyond x 4 in the series for H, 
j HHj dx =4wl 
There is no need to evaluate (3 and S for H, (xj — H, (—x : ) = 2x 1 (/3 -f- Sx 2 ), which 
immediately gives (3 + §<i if we put a** = 2/3. Greater accuracy may be obtained 
by suitably choosing a series of values of x, &c., to take into account the higher terms 
of the series, but the above expression is sufficient for the present purpose. 
The evaluation of the integrals by the above methods leads to the following table of 
values for u n in applying formula (53) to solenoidal coils of length 6 and radius a. 
Single Layer Solenoidal Coils. Radius = a. Length = b. u n in formula (53). 
bj2a = 0-0 0-2 0-4 0-6 0-8 1-0 
u n = 3-29 3-63 4-06 4-50 4-93 5-28 
The assumption u n = 3-29 -f- 6/a will give results which do not differ by more than 
2 per cent, from the above values. 
Flat Coils .—By methods similar in principle to those used in determining the values of 
u n for solenoidal coils, the following values of u n have been found :— 
Single Layer Flat Coils, r = inner radius. R = outer radius. 
rj R = 1-0 0-9 0-8 0-7 0-6 0-5 
u n = 3-29 3-36 3-58 3-84 4-24 4-78 
(15) Single Layer Coils at High Frequencies. —When z is greater than 3, F ( 2 ) and G ( 2 ) 
assume the simplified forms 
F(s) = (a /22 — 3)/4, G ( 2 ) = (y/ 22 —l)/8, 
so that formula (53) becomes 
R' = a + /3z, . (66) 
in which 
« = iR(2-«„d 2 /D 2 ), /3 = ^|L(2 + m^7D 3 ). 
Now u n d 2 / D 2 seldom exceeds 6, so that a /(3 will usually lie between -1-6*7 and — 0-4. 
