Mr Campbell, The study of discontinuous phenomena. 123 



the solution of the problem, interesting questions are raised in a 

 branch of mathematics which has received little attention. 



§ 5. Let the period of observation be divided up into m periods 



T 



T = —, such that/(^/) may be considered sensibly constant during 



any one period t. The total length of the period of observation T 

 (that is the time from the moment of insulating the electrode to 

 that of taking the observation) will be considered to be the same 

 for all observations : but from the form of /"(i/) it is clear that all 

 periods T may be considered equal so long as aT, ^T and pT are 

 all large compared with unity. 



Let yr be the number of particles emitted during the rth 

 period t. Then, during any one period of observation, yr is a 

 function of r. In this statement the word ' function ' is used in 

 the widest possible sense, and not in the restricted sense often 

 employed in analysis. Further, to each value of r during any one 

 period T, corresponds one and only one value of y,.. Let us write 



2/.=</>w (n 



The function <^(r) has different forms in different periods T. 

 Let there be v periods of observation and let the form of 4>(r) 

 in the pth period be cf)p (r). Since (j) (r) can have a large number 

 of forms, a definite meaning can be attached to the expression 

 ' the probability that ^ (r) has the form (f)p (r) ' : let ^p be this 

 probability. 



Let Nt + fl^i, Nt + X2, ... Nt + Xn be all the possible values of 

 y : then (f)p (r) has m of these values during the m periods into 

 which T is divided. The probability of the occurrence of any 

 value Xr is, by § 2, 





{Nt + X,) ! {NT -Nt- Xr) ! \T) \ T) 



Hence the probability that ^ (r) has a definite m of the possible 

 values of y, that is, the probability that <^ (r) has a definite 

 form, is 



%i-%2. •..%m = ^P (8). 



Now it is important to notice, for this is the essential step 

 in the argument, that the value of $p is not affected by the order 

 in time in which the definite values of x^ occur, ^p is the same 

 if the values of x^ are the same : it is not altered if x^ = a^ when 

 r = u and a^ when r = v, instead of a^ when r = u and a„ when 

 r = v. That is to say, (pp is wholly independent of r. 



Now if (fi(r) has the form <}>p(r), the corresponding value of 

 ^V will be 



^v = [s;:r</'pW/0'T)p (9). 



