540 
MR. J. S. TOWNSEND ON MAGNETIZATION OF LIQUIDS. 
rr (]>, ra rly 
(1.) 47t 2 r 2 - dr -J- 47r 2 r 2 — dr -f 27mm — 
v ' J 0 at J ,. at 
— 4tt 2 ^ 
0 nU 
Differentiating with respect to r, and dividing by r and differentiating a second 
time, we get 
d~u 1 rlu v, 47t clu 
dd r ar d a dt ^' 
Let u — ve m<rt , where v is a function of r only and m as yet undetermined. 
Hence we have : 
, d?v dv . o \ 
r l —r + r —- 4- (47rmr J — 1) -y = 0. 
ar- ar ' 
Hence 
v 
= 47 Tin r, 
where J 1 is Bessel’s internal function of 1st order. 
Hence 
u = %A m e~ m<rt J r v/47 Tin r, 
substituting in (1) and using the equations 
and 
we get 
JiO)=-^rJo(A 4. (^J.) = ^ 
x~ [Jo [a') —(— Jy (>r)[] — 2a’J yXd 
SA m crc~" l<rt ‘s/ 47rin r 2 7rJ 0 (v/47rm a) = 
47T 2 7’%E -tf 
L £ 
Hence J 0 \/inm a = 0, except when mcr = R/L = p , in which case 
v. 
The other coefficients are determined by the condition that u = 0, when t = 0. 
From what follows, however, it will be seen that it is unnecessary to determine them, 
as they do not appear in the result. 
The number of lines of force through the secondary due to the induced currents in 
the liquid is 
4tt 3 f r % u dr = 2B m e~ mtri + B /; e“ p< . 
J o 
* Forsyth’s ‘ Differential Equations.’ 
