450 Wisconsin Academy of Sciences, Arts, and Letters. 



If, however, we differentiate the formula three times with re- 

 Bpect to/> we have 



e ^ ^ x3 sin 2r^xdx =V_1_^—^ Ua*-""Tr'= J 



which we can employ in the reduction of the other pr4rt of the 

 integral ; thus 



Finally the equation 



V= V— V" [141 



shows the probability of an error lying between — cz and + cz ; 

 the second part V" either vanishes for large values of r or is 

 very small. 



§ 10. 



If Ay denotes the probable error, according to the definition 



y^f^ will equal %. In order to find li from this, first take F" 



= and compute h by means of the equation V'^, = ^/o. Thus 

 by the aid of a table w^hich shows the value of the integral 



corresponding to any value of z it is found that z = 0.47693(5 

 if the integral itself equals 1/2- This value of z being denoted 

 by Pi we have from the equation V'c = Yz 



