520 
PROFESSOR H. LAMB ON ELECTRICAL 
stand for a linear dimension of the conductor and p for its specific resistance, it will 
appear in the sequel that when as in all practical cases A is small compared with 
v/L, the error introduced by the assumption in question is of the order \pjv~. For 
any ordinary metallic conductor, and for any value of A which can be appreciated 
experimentally, this fraction is excessively minute. 
In § 3 the solutions of our equations (on the assumption above indicated) are given 
in the form appropriate to our present problem. These solutions are of two distinct 
types. Those of the first type, which are much the more important from an experi¬ 
mental point of view, have (I find) been discussed, though by a different method, by 
Professor C. Niven in a paper recently published.* As the points to which attention 
has been directed are for the most part sufficiently distinct in the two investigations, 
I have allowed the corresponding portions of my paper to stand. 
In § 4 I discuss the case of electric currents started anyhow in the sphere and left 
to themselves. The equation which gives the “ moduli ” of the natural modes of 
decay of the first type agrees with the result obtained by Professor Niven. 
In § 5 is studied the case of induced currents. Since any disturbance in the field 
(however arbitrary) can be expressed, as regards the time, by a series of simple 
harmonic terms, it is sufficient to consider the case when the variations in the 
inducing system follow the simple harmonic law. This case has moreover acquired 
a special interest since the invention of the telephone. 
The two extreme cases, when the period of the variation in the field is very large 
or very small in comparison with the time of decay of free currents in the sphere, are 
discussed in some detail. 
In § G the case of a thin spherical shell is briefly examined. 
I next proceed to investigate what modifications must be introduced into the 
methods and the results of the preceding sections when the substance of the sphere 
is susceptible of magnetisation. This occupies §§ 7, 8, 9, 10. 
In the remaining sections of the paper I investigate the solution of our fundamental 
equations, taking account of the finite value of v. The corrections to our former 
results are of most interest in the solutions of the second type. Although the 
preceding theory, based on the assumption v— co , is sufficient for all purposes of 
comparison with experiment, there are certain processes of (at all events) theoretical 
interest of which it fails altogether to give an account, viz,, all those cases in which 
any change in the superficial electrification of the sphere takes place. For the 
expression of these the solutions of the second type are appropriate. There is no 
difficulty in working out the requisite formulae, but in the application to the case 
of free motion a difficulty of interpretation arises which is noticed in the proper 
place. 
1. Let us suppose that we have one or more conductors at rest in an insulating 
* Phil. Trans., 1882. The date of the paper is January, 1880. 
