124 Mr Campbell, The study of discontinuous phenomena. 



The probability of this value of ^V is ^p : hence the sum of the 

 values of 6'^t for the v periods of observation is given by 



2.6V=^2J:;^,K:r0p(r)/(rT)P (10), 



where the first sign of summation denotes summation with respect 

 to all the forms which (r) takes in the v periods. 



§ G. Now summation with respect to a large number of forms 

 of a function is not a process of which the methods have been 

 elaborated. The value of the right-hand side of (10) cannot be 

 found by a direct method, but it can be found by means of the 

 following artifice. 



Expanding the squared factor in brackets we obtain 



(11)- 



Since <l>p is independent of r and s, we may reverse the order 

 of summation and write 



X. e-^T = V XIT KZi ^:=i ^p [N'r^f(rT)f{sr) 



+ Nt (wr + Xs)f {rT)f(sT) + XrxJirT)f{sr)'\ (12), 



where the inner summation denotes that we take the average 

 of the terms in square brackets over all periods of observation 

 before summing them for the periods into which a single period 

 of observation is divided. Now Nt is a constant and, as has been 

 emphasized already, the average value of Xy or Xg is independent 

 of r or s. The value of Xr in any one period T depends on the 

 Value of r, but the probability that it will have that value is 

 independent of r. Hence the average value of {x^ + x^ or x^-Xg 

 is independent of /(rr) /(sr) and we may take the latter factor 

 outside the summation with respect to p. Further we may notice 

 that the average value of Xj. or Xg is zero by § 2 : hence the average 

 of {Xr + Xg) is zero and the average value of x^Xg is zero, except 

 when r = s. (It must be remembered that we are not taking the 

 average for different values of r and s, but the average for the 

 same values in different periods, so that no ^^ is a member of Xx^.) 

 Hence our equation finally reduces to 



2.^v= V [s;:r K=x NH^f{rT)f{sT) + ^TJ^fivT) m\ ^.^A 



(IS). 



But Sp^)^ ^pXy? is simply the average value of x.,? for all periods T, 

 that is, the average of Xy- for a very large number {v) of cases 

 which are perfectly independent. By equation (1) above, the 

 value of this average is Nt and hence 



2. e^T = V ts;::r s::r iYv/(rT)/(.T) + s;:r Nt .p {vt)] . . .(14). 



