SECT. 5.] Arrangement and Formation of the Series. 29 



5. In support of an affirmative answer to the former 

 of these two questions, several different kinds of proof are, 

 or might be, offered. 



(I.) For one plan we may make a direct appeal to 

 experience, by collecting sets of statistics and observing 

 what is their law of distribution. As remarked above, this 

 has been done in a great variety of cases, and in some 

 instances to a very considerable extent, by Quetelet and 

 others. His researches have made it abundantly convincing 

 that many classes of things and processes, differing widely 

 in their nature and origin, do nevertheless appear to con 

 form with a considerable degree of accuracy to one and the 

 same 1 law. At least this is made plain for the more 



1 Commonly called the exponential 

 law; its equation being of the form 

 y = Ae~ hx *. The curve corresponding 

 to it cuts the axis of y at right 

 angles (expressing the fact that near 

 the mean there are a large num 

 ber of values approximately equal; 

 after a time it begins to slope away 

 rapidly towards the axis of a; (express 

 ing the fact that the results soon 

 begin to grow less common as we 

 recede from the mean) ; and the axis 



of x is an asymptote in both direc 

 tions (expressing the fact that no 

 magnitude, however remote from the 

 mean, is strictly impossible ; that is, 

 every deviation, however excessive, 

 will have to be encountered at length 

 within the range of a sufficiently long 

 experience). The curve is obviously 

 symmetrical, expressing the fact that 

 equal deviations from the mean, in 

 excess and in defect, tend to occur 

 equally often in the long run. 



A rough graphic representation of 

 the curve is given above. For the 

 benefit of those unfamiliar with 



K 



mathematics one or two brief remarks 

 may be here appended concerning 

 some of its properties. (1) It must 



