122 Mr Campbell, The study of discontinuous phenomena. 



applied.) Then the equation of motion of the needle subsequent 

 to the communication of a charge E is 



I^ + l^f^+ke-K.EIC.e-P^ = (2). 



Taking into account the initial conditions the solution is : . 



^ Ae-'^' + Be-P' + Pe-i" (3), 



where 



_ /x-\V-4//> - KE ^p^-P 



"" 21 ' C{If-fip + k)' a-yS' 



^~ 21 ' a-^' 



All these quantities can be found by suitable experiments 

 on the electrometer and the resistance. Accurate determinations 

 of them would require some ingenuity and labour, but they can 

 be found with accurac}^ sufficient for the purpose of this research 

 by methods which will be obvious to everyone. (See § 10.) 



It is easy to show that, subject to the assumptions njade, the 

 effects of charges communicated at any times, Tj, T3, ... T;„, are the 

 same as if the needle had been at zero and at rest at the moments 

 of communication. Hence the deflection at a time T, which is 

 subsequent to the communication of all the m charges, will be 

 given by 



&T=K^Jl (^^"'"'^""••' + 5e-^(2^-^'-t +Pe-i'(r-v)) (4)_ 



Let us now change the origin of time to the time T, which 

 is the time of observation. Writing 1,.' for the time before the 

 moment of observation at which the rth particle was emitted, 

 we have 



Ot - ^721 {Ae-'^'r + Be-^'' + Pe-^"0 (5) 



= KV:f{tr){^^j) (6). 



6t is a function both of the number of particles emitted and 

 the times at which they were emitted. In the case which w^e are 

 considering we do not know these times and hence cannot determine 

 the number directly from a single observation. But I shall proceed 

 to show that we can determine it from the average of a large 

 number of observed values of ^V- (It will appear later why we 

 have to take the average of 6'^x find not of l^r|.) For the main 

 principle of the argument I am indebted to Mr G. H. Hardy, 

 Fellow and Mathematical Lecturer of Trinity College, Cambridge, 

 who points out that though no elaborate analysis is necessary for 



