IN CYLINDRICAL CONDUCTORS, ETC. 
63 
and when n = 0 
^(xo-x'o) = ~2+W i z )- 
Using these expressions in the equation for W with (B n ) for the modulus of B,„ 
+ (B n ) 2 a 2 n \fr n (z) 
1 
( 16 ) 
The energy dissipation in unit length of the cylinder is thus expressed in terms of 
coefficients B n depending on the form of the applied magnetic field and of functions 
\[ s n having argument z — 2a (tt&co) 4 - The functions <p n , \\ r n are discussed in the next 
section. As regards the coefficients B„, if the components of the magnetic force in the 
undisturbed field are P 0 , Q 0 these components may be expressed in the forms 
P,, = 2K/ 1 sin(w6 + a Ji ) 
i 
Q 0 = — + 2 K n r n_1 cos (nd + a n ) 
r i 
at all points outside the cylinder as these expressions are derivatives of a potential 
function satisfying Laplace’s equation and constant along the axis of the cylinder. 
Further, by differentiation of (8a), similar expressions to (17) are obtained, when 
X is made zero—that is, when the disturbance due to eddy-currents in the cylinder is 
removed. These expressions are identical with (17) if we make 
Bo : iw K ( 
Oj 
B n = iwK n /n. 
Hence, using K„ in place of B n in (16) 
W = L 
4ft) 
(K„F U +bh(z)\ + 2(K.)*«V. (*)/» 
( 18 ) 
(4) The Functions <p n mid \t, n . —Tliese functions are defined by 
<p n (z)-i\[r n (z) = J /i+] ! W-iz). 
Series formulae for these functions have been developed by the author.* 
The cases n = 1 and n — 2 are the most important ones, and in these cases <f> and \Jr 
may be expressed in terms of ber and bei functions as follows :—- 
Let 
X (z) — ber 2 z + bei 2 0 
Y ( 2 ) = ber' 3 z + bei' 2 z 
Z(z) = ber z ber' z + bei z bei' z 
W ( 2 ) = ber z bei' z —bei z ber' z 
( 19 ) 
* Butterworth, ‘ Proc. Phys. Soc. Lond.,’ vol. XXV, p. 294, 1913. 
