nr.riATiox or />■ i.v/'ou v n/z'/./r.? m 



\ii\\ .i-^unu- lli.it 



is lU'uligihli' rnmp.irtil with 



and also assimu- that ?, c and (m — f) arc such that a|)pr<)ximati'K- 



Jx^(l-x)'-'dx = J x-'(l-x)"-'dx. 

 1 lu-ii, tiiially. 



£\x'{l-x)''-'dx 

 / .v-(l-.v)— rf.v ^■^" '>■'''' 



This well known formula niii^lit ha\e i)t'en obtained by assuming 

 ab initio that 11' (.v) is independent of x. It should be particularly 

 noted that this independence is not identical with the assumptions 

 made alx)ve. In the applications which are here contemplated the 

 values of pi, c and n are such that g need be but a small fraction of 

 the range o to 1. 



In the "Theorie AnaKticiuc" Laplace transforms (3) so that it 

 can be evaluated in terms of the Laplace-Bernoulli integral 



-4- re-"di, 



V 



where k is a function of pi, c anfl n. This transformation is most 

 valuable when pi is in the neighborhood of 1/2. For small values of 

 pi the transformation which converts the binomial expansion to 

 Poissf)n's exponential binomial limit is more appropriate and gives, 

 writing (w pi) =ai, 



P = ~J""ye->dY = P{c + \,ai). (4) 



