1883.] On Electrical Motions in a Spherical Conductor. 131 



beforehand from the researches of Helmholtz* and others that ths 

 results (so far as they are peculiar to the theory) would be of far too 

 subtle a character to admit of comparison with experiment. In 

 studying the mathematical character of the problems above stated I 

 was led to a certain system of formulas, which I have since utilised in 

 two communications to the London Mathematical Society, "f and which 

 seem likely to be of use in a great variety of physical questions. 



The first section consists mainly of a recital of the fundamental 

 equations and of the conditions to be satisfied at the surface of a 

 conductor. It is assumed, in the first instance, that the magnetic 

 susceptibility of the conductor is zero. 



In § 2 is introduced the assumption that all our functions vary as 

 e kt , where t is the time, and \ a constant. It is pointed out that this 

 assumption is sufficiently general. The fundamental equations are 

 then put into a mathematically convenient form. Before, however, 

 proceeding to apply these equations as they stand, I examine the effect 

 of assuming that the velocity (v) of propagation of electro-magnetic 

 effects in the medium surrounding the conductor is practically infinite. 

 This assumption, which has been made by all writers (including 

 Maxwell himself) who have applied Maxwell's theory to ordinary 

 electro-magnetic phenomena, greatly simplifies the calculations with- 

 out sensibly impairing the practical value of the results. If L stand 

 for a linear dimension of the conductor and p for its specific resist- 

 ance, it will appear in the sequel that when, as in all practical cases, 

 X is small as compared with v/h, the error introduced by the assump- 

 tion in question is of the order Xp/v^. For any ordinary metallic 

 conductor, and for any value of X, which can be appreciated experi- 

 mentally, 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 experimental 

 point of view, have (I find) been discussed, though by a different 

 method, by Professor C. Niven, in a paper recently published. J 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 JNTiven. 



* Crelle, t. 72 (1870). 



f " On the Oscillations of a Viscous Spheroid," " Proc. L. M. S.," Xor. 10, 188J, 

 and " On the "Vibrations of an Electric Sphere," May 11, 1882. 

 X " Phil. Trans., ' 1882. The date of the paper is January, 1880. 



K 2 



