126 Mr Campbell, The study of discontinuous phenomena. 



In passing it may be noted that the argument cannot be 

 applied to the determination of |^r|. For, in the place of the 

 square of the expression in brackets in (10) we should have the 

 modulus of that expression. The modulus cannot be expanded 

 in powers of the variables concerned and, accordingly, the order 

 of the summations cannot be reversed. But on this reversal hangs 

 the whole argument. In any case it would be advisable to avoid 

 making use of | ^r | > for we have seen in § 2 that its value can 

 be found with accuracy only if Nr is very large. 



§ 8. Before proceeding to discuss in detail the application 

 of this theory to experiment we must consider how far the 

 assumptions that have been made are likely to be fulfilled in 

 experimental conditions. 



(1) We have assumed that r is small compared with T. 

 Since we have seen that the theory is true whatever the value 

 of Nt, it is clear that this assumption will cause no difficulty. 

 We have already, in replacing the summations of the last 

 paragraph by integrals, made t infinitesimal in comparison with 

 the time constants of the instrument. But T is the whole period 

 over which the physical conditions are constant, and it is clear 

 that the experiment will be arranged so that these are constant 

 over a period long compared with the time of observation. 



(2) (1) is the only condition which is essential to the theory, 

 but we have seen that in order to get a probable error less than 

 1 °/q, we must take a very large number of observations. The 

 quantity v corresponds to the cr of § 2 and hence, for this 

 degree of accuracy must be not less than 10*. Now it is clear 

 that 10* observations cannot be made by any process of looking 

 at the instrument and writing the observation down on paper: 

 on a favourable estimate the process would need a month's 

 continuous observation, day and night. Accordingly we must 

 start our discussion of what instruments are to be used with 

 considering how this large number of observations is to be taken. 



§ 9. The obvious method of taking this large number of 

 observations is to record photographically the fluctuations of the 

 instrument and to deduce the value of 6'^t from the resulting 



1 r^ 



trace by finding the value of ™ I 6''^j'dt. It would not be hard 



to devise a mechanical integrator which would give the value of 



I y^dx for any irregular curve, and, if this instrument were sensitive, 



its indications would correspond to an average taken over a very 

 large number of cases : in fact, if dx is the smallest interval over 

 which the instrument can be expected to distinguish between 

 different values of y, then an integration over a length x would 



