18 On certain kinds of Groups or Series. [CHAP. I. 



small scale, utterly irregular; it becomes however regular 

 when the numbers examined are on a larger scale ; but if we 

 continued our observation for a very great length of time, or 

 over a very great extent of country, we should find this 

 regularity itself changing in an irregular way. The sub 

 stitution just mentioned is really equivalent to saying, Let 

 us assume that the regularity is fixed and permanent. It is 

 making a hypothesis which may not be altogether consistent 

 with fact, but which is forced upon us for the purpose of, 

 securing precision of statement and definition. 



14. The full meaning and bearing of such a substi 

 tution will only become apparent in some of the subsequent 

 chapters, but it may be pointed out at once that it is in this 

 way only that we can with perfect strictness introduce thei 

 notion of a limit into our account of the matter, at anyi 

 rate in reference to many of the applications of the subject 

 to purely statistical enquiries. We say that a certain pro 

 portion begins to prevail among the events in the long run ;i 

 but then on looking closer at the facts we find that we have 

 to express ourselves hypothetically, and to say that if present) 

 circumstances remain as they are, the long run will show i1 

 characteristics without disturbance. When, as is often th( 

 case, we know nothing accurately of the circumstances bj 

 which the succession of events is brought about, but hav&amp;lt; 

 strong reasons to suspect that these circumstances are likely 

 to undergo some change, there is really nothing else to 

 done. We can only introduce the conception of a limit, to-| 

 wards which the numbers are tending, by assuming thai 

 these circumstances do not change ; in other words, by sub 

 stituting a series with a fixed uniformity for the actual om 

 with the varying uniformity 1 . 



1 The mathematician may illustrate analogies of the circle of curvature 

 the nature of this substitution by the in geometry, and the instantaneoi 



