32 Arrangement and Formation of the Series. [CHAP. n. 



the case of the extreme values, will become obvious on 

 a little consideration. In some of the classes of things to 

 which the law is supposed to apply, for example, the suc 

 cessions of heads and tails in the throws of a penny, there is 

 no limit to the magnitude of the fluctuations which may and 

 will occur. Postulate as long a succession of heads or of tails 

 as we please, and if we could only live and toss long enough 

 for it we should succeed in getting it at length. In other 

 cases, including many of the applications of Probability 

 to natural phenomena, there can hardly fail to be such 

 limits. Deviations exceeding a certain range may not be 

 merely improbable, that is of very rare occurrence, but they 

 may often from the nature of the case be actually impos 

 sible. And even when they are not actually impossible 

 may frequently appear on examination that they are onl) 

 rendered possible by the occasional introduction of agencie 

 which are not supposed to be available in the productio: 

 of the more ordinary or intermediate values. When, fo 

 instance, we are making observations with any kind c 

 instrument, the nature of its construction may put a 

 absolute limit upon the possible amount of error. And eve 

 if there be not an absolute limit under all kinds of usag 

 it may nevertheless be the case that there is one unde 

 fair and proper usage; it being the case that only whe 

 the instrument is designedly or carelessly tampered with wi 

 any new causes of divergence be introduced which were no 

 confined within the old limits. 



Suppose, for instance, that a man is firing at a marl 

 His worst shots must be supposed to be brought about 

 a combination of such causes as were acting, or prepare 

 to act, in every other case ; the extreme instance of whs 

 we may thus term fair usage being when a number &amp;lt; 

 distinct causes have happened to conspire together so 



