Mr Campbell, The study of discontinuous phenomena. 133 



§ 14. Let the suffix + denote in all cases quantities corre- 

 sponding to one source, and the suffix — denote quantities corre- 

 sponding to the other source. Then, it is clear that (10) may be 

 replaced by 



where ^+p (r) and <^_p (r) may have any of the possible values 



Nt-Vx^^,...,Nt + x^n, or -NT + x^^,...,-NT-\-x+n respectively, 



for we must remember that Nj^ = — N_. 



Reasoning in precisely the same way as before, we arrive at 

 the following equation corresponding to (13), 



s, {d^T + e_Ty = Kli F' (^^) sj=i ^+p ^-p (^+'- + ^-)' (28). 



Now 'fZ^ ^+p ^-p {x+r + ^-r)^ is the average value of {x^^ — x^rY 

 for a very large number of values of that quantity. But by a 

 well-known theorem in probability, the average value of {a + 6)^ 

 is a? + 6^ if a and h are independent. But the average of a?+/ 

 and of x_r^ is the same and equal, by (1), to INt. Hence, we get 

 in place of (15), 



^t.{d^T-^0_Tf = ^Nr^dtF"^{t) (29) 



or Il'^ = 2N{uT+v){sd.y) ' (31). 



Now, since A'^ is dependent on T, we cannot compare directly 



values of A' for different values of T. That is to say, if we record 



photographically, as suggested in § 10, the values of A'^ for all 



values of T, we cannot put 2iV^oc (average of all these values 



of A'^). We must compare values of A'^ for different values of T 



by dividing each value of A'^ by the appropriate value of {uT -\- v), 



A'^ 



and then take the average of —f=^ . We cannot even, as might 



° uT+v 



appear at first sight, take the difference of values of A' for a series 



of times differing by T and equate the average of the squares of 



these differences to 2iV {uT + v) : for such a procedure would involve 



the false proposition that 



|A/-A/|^ = A/2-A%. 



Accordingly the labour of deducing the value of iV from the 

 observed fluctuations will be very much greater than in Meyer 

 and Regener's method, for it would be difficult to construct a 



