456 ^Yisconsin Acadeiyuj of Sciences, Arts, and Letters. 



The approximate value of the mean error, Avhich we have 

 found to equal 1.032G6 by the exact formula [7] of § 7, will be 



6'^ 



^^'v uA^^^o^ 



from the formula of § 10 or, if the values found above are sub- 

 Btituted; 



To 



/ 



V 



'671 



1+^1, =1,03203, 



A more exact agreement could not be ashed. 



Furthermore it is seen that if y='20 the terms which depciid 



on fi' or on the fourth power of a are negligible in comparison 

 with the firs-t term. But since, as will appear below, the quan- 

 tity y is usually still larger or in a minor computation is not 

 far diuerent from that value, all the terms depending on (?' can 

 be neglect€xl altoirethcr without fear that the result will varj 

 far from the truih. So we can assume 



e-z-dz [17] 



as the probability of an error between the limits ±c^, 



'^y^P-\/ g- will be the probable error, [18] 



and 2x(? -i / -^ the mean error. [19] ■ 



Furthermore, if r = % expressed in terms of the last deci- 

 mal place, and we put C=c;^, then 



. /I 

 V ^ 



