SECT. 13.] On certain kinds of Groups or Series. 17 



towards a fixed numerical proportion in its uniformity; in 

 [the other the uniformity is found at last to fluctuate, and to 

 fluctuate, it may be, in a manner utterly irreducible to rule. 

 As this chapter is intended to be little more than ex 

 planatory and illustrative of the foundations of the science, 

 the remark may be made here (for which subsequent justi 

 fication will be offered) that it is in the case of series of the 

 former kind only that we are able to make anything which 

 can be interpreted into strict scientific inferences. We shall 

 be able however in a general way to see the kind and extent 

 of error that would be committed if, in any example, we were 

 to substitute an imaginary series of the former kind for any 

 actual series of the latter kind which experience may present 

 to us. The two series are of course to be as alike as possible 

 in all respects, except that the variable uniformity has been 

 replaced by a fixed one. The difference then between them 

 would not appear in the initial stage, for in that stage the 

 distinctive characteristics of the series of Probability are not 

 apparent ; all is there irregularity, and it would be as im 

 possible to show that they were alike as that they were 

 different ; we can only say generally that each shows the 

 same kind of irregularity. Nor would it appear in the next 

 subsequent stage, for the real variability of the uniformity 

 has not for some time scope to make itself perceived. It 

 would only be in what we have called the ultimate stage, 

 when we suppose the series to extend for a very long time, 

 that the difference would begin to make itself felt 1 . The 

 proportion of persons, for example, who die each year at the 

 age of six months is, when the numbers examined are on a 



1 We might express it thus : a it takes a very great number to esta- 



few instances are not sufficient to blish that a change is taking place in 



display a law at all ; a considerable the law. 

 number will suffice to display it ; but 



v. 2 



