2 Proceedings of the Eoyal Irish Academy. 



They consist of infinite series the first term of which is in each case the 

 corresponding term for the fixed sphere. The solution obtained here 

 consists of a method of approximating to as many terms as we please to the 

 surface-density arranged in descending powers of c, the velocity of radiation. 

 Quaternion notation and electromagnetic or electrostatic units are employed. 

 The latter units are much more convenient for this problem and generally 

 in electronic investigations than the " rational " units employed by Lorentz 

 and Hea\-iside. The most remai-kable results obtained are as follows : — If a 

 sphere be placed in a uniform field of force, and if it possesses no Newtonian 

 mass, it will move so as to have a uniform surface-density. If, however, it 

 possesses this mass, an excess of negative electricity is formed on the side 

 opposite to the eicceleration, and excess of positive on the opposite side, 

 following the simple cosine law in addition to the uniform layer. As the 

 mass increases, the cosine layers approach the electrostatic value for a fixed 

 sphere, but in all cases the total masses longitudinal and transverse have the 

 seime values as for a sphere having a Kni/orm fixed surface-density. It is also 

 true that the radiation is the same in both cases. Now the rigid electron of 

 Abraham* is the conception which agrees most closely with the now classic 

 experiments of Kaufmann, so that we see now that we may, if we please, 

 consider the electron to have the properties of a perfect conductor, or, if the 

 sphere has no mass, the interior might be an insulator. Phenomena such as 

 Rontgen rays might then be due to the oscillations on the electron itself. 

 Tlie mechanism might be such that the oscillations would not be so highly 

 damped as for those of a spherical conductor. Electrons of equal charge 

 might then differ from one another on account of being in different modes 

 of vibration. Certain phenomena such as the number of molecules ionized 

 by Rontgen rays and the differences in secondary /3-particles observed by 

 McClelland and others might be thus explained. The result obtained in 5 — 

 that a slow velocity diminishes the damping factor and lengthens the period — 

 would perhaps strengthen this supposition. 



II. — The Elkctkomagxbtic Equations axd the Boundaby Conditions. 



In free aether the electric force c and tlie magnetic force t\ are derived 

 from a scalar potential P and a vector potential 53 by means of the equations 



r't = - VP - ti, (1) 



c-'l = I'Vo, (2) 



where c is the speed of radiation and the units are electromagnetic. P and w 



" Prof. E. T. Whiitakcr points out that the defonnablc electron of Bucherer gives a better 

 •gieement with recent experimenlal results. 



