452 Vfisconsin Academy of Sciences, Arts, and Letters. 



as the probability of an error cx . Tben by putting x^X^ = z 



and cos ex instead of e~^^^ (since the im:i^inary j^a-rt is can- 

 celled) we shall have 



01^ \ ^ H 





Pinally if we employ the reduction formulas 



e cospzaz = « / e 



and its fourth derivative 



/5= 



2a V4^4 4^8 ^I6a8/ 



we shall have 



/A 3c» 



2(r 



( ^ / 3 _ 9c« , 9c« ^ ) 



as the probability of an error Cy, 



The integral of this being taken between the limits — c and 



+ c, in which - takes the place of dc, it is again necessary to 



avoid the avoid the formula for V as found in § 9. This is 

 shown by the computation. The partial integration can be per- 

 formed according to the formula 



I e-63^22n+2d2=?!^hl I , 



e-bz^z'i.ndz— —e-bz^z2n+l 



