128 Mr CampheU, The study of discontinuous phenomena. 



If the quantities a, ^, p are all different, it will be no easy 

 matter to find their separate values: the analysis of a curve into 

 three component exponentials involves great labour and the result 

 is not likely to attain a high degree of accuracy. But most 

 electrometers and electroscopes are either just periodic or just 

 aperiodic, and they can be adjusted to the boundary condition 

 between these two states without great difficulty. But in this 

 boundary state 



V/A^ _ 4,1k = and a = /8. 



Let us suppose, then, that /3 = a + 7, where 7 is small compared 

 to a. Taking into account the values of the constants, A, B,A', B', 

 our equations undergo the following simplifications. (23) becomes 



61 = ( F - P') e--^* + Fe-v^ - rytB'e-'^' (23'), 



f{t) (equation (6)) reduces to 



f(t) = P (e-i'* - e-«) - r^tBe-'^' (6'), 



and (21 ) becomes 



p./l .1 2 \ /7/3 7/3 ^ , 7^^- 



,, 1 ^ \2p'^2a a + p)'^ Via? {a + pfJ^ 4«^ , , 



From (6') and (21') it is evident that nearly equal values 

 of p and a are to be avoided: for, if these two quantities are 

 equal, the two equations contain only terms involving the small 

 quantity 7, which is hard to determine. Accordingly we must 

 make one of the two quantities p and a very large and the other 

 very small. There are three reasons why a should be made large 

 and p small. 



(1) A small value of p corresponds to a high value of the 

 resistance of the Bronson cell. But the greater this resistance, 

 the greater is the deflection of the indicating instrument for a 

 given current passing through the cell. Since it is easier to 

 measure variations of the same proportional amount in a quantity, 

 when that quantity is large, than when it is small, it is desirable 

 that the steady deflection of the indicator, corresponding to the 

 mean current, should be as great as possible. 



(2) A large value of a makes the terms involving the 

 unknown small quantity 7 negligible. Any slight deviation from 

 the boundary state between periodicity and aperiodicity will be 

 unimportant. 



(3) If a is large the required constants can be determined 

 more easily from the equation (23'). Since p and a. are very 

 different, the value of the terms involving e~** can be found from 



