Bremiher — Em^ors Affecting Lonanthmic Computations. 433 



§ 4. 



If r denotes the extreme limit of the error by which the log- 

 arithms of a given table can be affected, all the values which / 

 can have are shown by the series 



n n—1 n — 2 ^ , n — 1 , n 



in which n is an infinite number. The total number of 

 tliese values is ^n + 1. And it is a necessary assumption that 

 each one of these values will occue. once, and only once, since 

 there is no cause at vv^ork which will give predominance to one 

 value rather than another. 



Then if we form a series of the values of a^f^, this series, 

 which is merely the previous one multiplied by a^, will be 



n n—1 n — 2 ^ , n — 1 , n 



•";r^^^' — ^^^^'--^-^i^--- ^••••+-^^^^' + ;r^^- 



We shall have a similar series for eachn/; the number of terms 

 is the same in all, but the intervals are not the same. In place 

 of these, and for the sake of simplicity, let the various series be 

 so written that the terms have a common interval. It is evi- 

 dent that this is legitimate, since it is merely so distributing the 

 total interval between — (x-^y and 4- oc^y that the number of small 

 intervals is changed. Thus instead of ^i/i, a-j2^ .... we write 

 the series 



n n ' n n 



in which the difference between the terms is everywhere the 

 same, namely, — » but the number of terms is respectively 



2n<2:i+l> 2na^-\-l.. .2na +1. 



v 



