136 Mr. 0. Heaviside on the 



where M and N are the following functions, 



oJ' ' 



(37) 



i standing for v — 1, and^? for s/kiry^ktfi* Also 



P=M + ^M / , QssN + gN', . . . (38) 



the ' denoting differentiation to r. In (36) M and N have 

 the values at distance r, and P , Q the values at r=a ly the 

 boundary. 

 We have 



P 2 + Q 2 = M 2 + W + 2q(MW + NN') + ? 2 (M /2 + N' 2 ). (39) 

 If ?/=(^ 4 =(47r/i 1 A; 1 r 2 w) 2 , we have the following series : — 



+ JN - i+ 2.4 2 V i + 2 6.8 2 l i + 3 10.12 2 l 1 + 4 14.16V + • , • , 



+ iN " 4r 2 l 1+ 4 2 6 2 r + 8 2 10 2 l i+ 12 2 14 2 V 1+ 16 2 18 2 V +'"' 

 MM' + NN'--^- (l I 6y fli 5 ^ fl i- 4 ^ (\ ■ 4 ^ fi l 



MM +JNJN - 16 ^ ^+ 6 2 8 2^+ 10 2 12 2^+ 14 2 16 2^+ 18 2 20 2^+--- V 



These are suitable for calculating the amplitude of T or of C 

 when y is not a very large quantity. The wire current C is 

 given by 



c= ~2^lw^r m r- tan py-QM' ) (41) 



where P, Q, M, N, M', N' have the boundary values. As for 

 M and N themselves, their expansions are 



M-i y \ y 2 i 



2 2 4 2 + 2 2 4 2 6 2 8 2 '"' I (42) 



~ 2 2 2 2 4 2 6 2 2 2 4 2 6 2 8 2 10 2 * ' J 



But these series are quite unsuitable when y is very large. 

 Then use the approximate formulae 



J °^Hi)* C0S ( sr -j)> 1 



T, x /2 \ 4 / Ml"' (48) 



(4 



