444 Wisconsin Academy of Sciences, Arts, and Letters, 



erroKS, computed fram the errors without regard to sign. We 

 obtain it if we multiply the error (5 — 2 m) r by its probability 



and double the integral of the product taken between the limits 

 and Y2 5. By this integration, either direct or as in § 5, we 

 have 



(^-M;:^!^..-^. |(i)'-+i-^(|-..ar+i+^(|-..«)-+i-.. j [17, 



In this formula we are to take only those sums of powers that 

 have positive exj)onents. If we take the same example as in § 6, 

 "we have, according to this fonnula, the mean error equal to 

 1.03266. The probable error, however, is found to be slightly 

 less than 1. 



§ 8. 



By the means proposed above we can determine the probabil- 

 ity of an error within any assigned limits in any quantity com- 

 puted by logarithms. Since, moreover, we have already shown 

 how the probable error and the mean error can be found, the 

 general problem may be cc>nsidcred as solved; unless there re- 

 main further difficulties in applying the formulas. For it hap- 

 pens that, whenever ^ is a large number, the separate terms of 

 the series employed are formed from very large numbers which 

 in turn cancel each other, so that it is necessarj^ to use many 

 more decimal places in the computation than are required in 

 the completed result. Thus in the example in § 6, eight place 

 logarithms were used in order that the probability mic:ht be 

 computed to four places. And for greater values of v the dif- 

 ficulties are increased to such an extent that if r = 100 the 

 computation of the values can scarcely be undertaken. 



In order to avoid these difficulties, theoretical accuracv must 

 be to some extent sacrificed and resort had to approximate 

 values, in order that we may express results in terms of the 



integral of e , which is of fundamental importance in discus- 

 Bions of probability. 



