
an Alternating Circular Electric Current. 339 
or in the equivalent form 
n° C*xaV ( {Joe—2 (22) + Jo¢42(22) ae 202n "a7 J og( 2x) }dax, 
e'o : 
the latter of which can be expressed also as 
2 
m2C*xaV {J ak 2xa)+(1— ok "a ”) { Jog(2v)dz}. 
0 
This expression agrees with that obtainable from Pock- 
lington’s paper * by using the relation there obtained between 
the current and the electric force at the surface of the wire. 
(The solution there taken appears, however, to correspond to 
divergent, not convergent, waves.) 
4. In case xa=o the rate of radiation thus assumes the 
simple form 
172C2a VJ3,(20). 
5. When c=0 the expression obtained for the general 
case in § 3 has, as previously stated, to be doubled, and the 
result may thus be written 
K 
27?C*xaV ( Jo(2x)da. 
~0 
In case xa is small this becomes 7?C?«4a*V/3. The radiation 
in this case was investigated by FitzGerald +; his result is 
one-half of the above ; he has, however, taken the mean value 
of sin? (colatitude) over a sphere as 4 instead of 2. 
6. If xa is very great we may use approximate forms 
of the hypergeometric series for large values of the variable, 
or may proceed mere simply as follows :—Except for small 
values of 6, the 2 component of the magnetic force at a great 
distance is now very small compared with the resultant of 8, y, 
and the first formula for the mean rate of radiation assumes 
the approximate form 
1/2 
1? 2K2a2V ( sin 63; («a sin 0)d0. 
=70 
Evidently in this integral we may use the approximate 
formula for J{(z) when z is large, viz.: 
(2/mx)? sin { (26 +1)x/4—2}, 
* “Electrical Oscillations in Wires,” Proc, Camb. Phil. Soc. 1897. 
+ ‘Scientific Writings,’ p. 125; Trans. R. D. 8S. Nov. 18, 1883. 
