440 Wisconsin Acadcray of Sciences, Arts, and Leiters. 

 The value of this will be equal to unity if we put 



representing thus the certainty that the error must always be- 

 between 



If, however, we had preferred to develop the series in § 4 ac- 

 cording to decreasing j)owers instead of increasing, Ave should 

 have reached the same formula for determining the probability 

 of an error situated between 



(ariH-a'a-f-....4-ay )r and {a^-\-a^-\- ....-\ra^ —2m)y 



And if we put 



m = i(^i-l-^2+-.--+^y ) 



it must be assumed that the value of this equals %, as indeed is 

 the case. But if we jDut 



the formula will still give a value equal to unity; this and many 

 other properties of the same formula, which can be proved by the 

 theorv of finite sums and differences, it is needless to follow 

 out here. 



Lastly, if we designate the probability [6] found above as 



then J Vm is the probability that the error lies between _ sy ^^<^ 

 — (5— 2m)x ^^d y2 — Wm is the probability that it is between 

 —(."?— 2 -n);^ and 0. And so, since the probability is the same if 

 the limits are positive, the probability that the error lies between 

 the limits ± {g—%n)y is l—IWm ? ^^ which m- receives in order all 

 the values from to ]/2S. 



