530 TRANSACTIONS OP SECTION A. 



Is (2''— 2) divisible by p 1 [p a prime] ? 



It is believed that no case is known of 2" — 2 = (mod. p' 1 ) with p prime. It is 

 staled by M. F. Proth l that 2''— 2^0 (mod. p"-), with p prime; but no proof is 

 given, no proof is quoted, no statement made of existence of any proof. At present 

 it can only be asserted that it is probably true. To test this the writer has tried all 

 prime divisors pit 1000 and finds that 2 — 2=£0 (mod. p 1 ) up to that limit. 



6. The Initial Motions of Electrified Spheres. 

 By J. W. Nicholson, M.A., D.Sc. 



The problem of the initial motions of electrified spheres has been treated recently 

 by G. W. Walker, 2 and the present paper consists mainly of an examination of some 

 interesting cases, with corrections of detail. It discusses the motion of a sphere, 

 conducting or dielectric, whose mass is purely of electrical origin, without a New- 

 tonian element. For a conducting sphere of radius a and charge e, with electrical 

 mass m' and small Newtonian mass m, starting from rest at tf = in a uniform field 

 F of electric force, the displacement of the centre at time t is 



2m'\ o 3 cV 3»t'c 2 I a\m J 2\m'J . a\m ) J 



and the surface density is given by 



1 e -^ ~- ct ctfm'W 



4ir<r = - + Ye a« cos ( — ) " cos 6. 

 a- a\m/ 



The corresponding formulas for a small mechanical force G are 



G ( fS + 2at 2a*\_Qa*^ ( QS ct,vi\\ 3/» \J gin «(«<*), 

 * 2m'\ c c*J w'c' I a\m ) 2\m' J a\m ) J 



, e 3G |' "-';' at/m'\l I . 



\ita— — . l-e« cos ( — I \ cos 6. 



or e v a \m J J 



The difficulties connected with the limiting forms of these expressions when m is 

 zero are discussed, and it appears impossible to ascribe an initial acceleration to the 

 sphere without introducing imperfection in the conductivity, although the electrical 

 distribution on the sphere tends to become uniform very rapidly. These results have 

 a bearing on a possible conception of the electron. 



The theory of the corresponding problem for an insulating sphere is also traced, 

 and it is shown that these special difficulties are absent, and that a simple solution 

 may be obtained when the dielectric constant is not small. The vibrations initially 

 set up in this case have a much smaller rate of dissipation, but many possible periods 

 instead of the single one belonging to the conductor. 



A discussion of these periods and rates of decay is given for special values of the 

 ratio iii'im. When this ratio is zero, or the sphere fixed or uncharged, the question 

 has been treated by Lamb, 3 an additional period with a rapid rate of decay having 

 since been indicated by Walker. The additional period is not present when in = 0. 

 For a rigid or uncharged sphere, the fundamental mode (ignoring the extra mode) is 



found to have a dissipation factor «-** where m— „ °, P = 4-493. 



k 3 a 



When m - 0, n = r —-. 

 k 2 a 



For a value of k=5 . 10 6 , and a sphere of molecular size (a = 1-3 . 10~ 8 ), as in 



Lamb's model of a molecule exhibiting selective absorption of light, the values of n 



become respectively T5 and 1-9 . 10 6 , so that, while the vibrations in the former 



case may be fairly permanent, those in the latter, for a sphere of the same size, and 



dielectric coefficient, will decay rapidly, and the dielectric sphere rapidly assumes 



a constant acceleration in a weak uniform field. 



1 Coinptes Rendu s des Seances de V Acad, des Sciences, Paris, t. 83, p. 1288 

 4 Roy. Sue. Proc, a77, p. 260 ; Phil. Trans., A 210, p. 145. 

 3 Canib. Phil. Trans., Stokes Commem. Volume. 



