530 
PROFESSOR H. LAMB ON ELECTRICAL 
surrounding the sphere but right away to infinity. Hence we must have, in (32), 
X,j = 0 ; and thence, by (37), 
i/r„_i(£R)=0.(47). 
The roots of this equation in kR are ail real. For the case n=l we have 
"When n— 2, 
When n=3, 
JcR/n=l, 2, 3, &c. 
kR/n= 1-4303, 2-4590, 3'4709, &c.(48). 
AH/tt= 1-8346, 2-8950, 3'9225, &c.(43). 
When the value of k for any particular mode is known, the corresponding value of 
X is given by (20). If r denote the modulus of decay, i.e., the time in which the 
currents fall to 1/e of their original strength, we have 
__ //jR\- 2 Pd 
7T \ 7T 
(50). 
For any given mode r is proportional to the square of the radius, and inversely 
proportional to the specific resistance; a result which may easily be obtained 
otherwise, by the method of “ dimensions.” 
For a sphere of copper [p=1642, C.G.S.] the modulus of the slowest mode of 
decay is 
T=-000775lt 3 second, 
the unit of It being the centimetre. For a copper sphere, of the size of the earth 
[R=6'37 X 10 8 ] the corresponding value of r is very nearly 10,000,000 years. 
As regards the nature of the various modes we may observe that the lines of flow 
of electricity inside the sphere are the intersections of the spheres r= const, with the 
cones y„/r" = const.; in other words, they are the contour lines of the harmonics on 
a series of spherical surfaces concentric with the origin. The intensity of the current 
at any point is proportional to r//„(^r).cZy„/de, when cle is an elementary angle at the 
centre of the sphere in a plane perpendicular to the line of flow passing through the 
point in question. The direction of the flow changes sign as we cross either the 
spheres for which \jj /t (kr) = 0, or the cones for which dy„/c/e = 0. The components of 
the magnetic induction at points outside the sphere are, by (35) 
a=nR*-+V#E)£x«’- _s '' 1 ' 
b=nR^UkR)^ x> ,r-^-' J-. 
c=«R*»+V.(*®)|x^' 8 *" 1 
(51). 
