44 BELL SYSTEM TECHNICAL JOURNAL 



Due to the reciprocal relationship between n and X, we may obtain 

 in a similar manner 



("T')^7^i^=("t')(^)'('-^)"^'"'--^^-) 



when t = {XlN)n. 



It is by means of these relationships that we have calculated the 

 cases for N > 1,000 as shown on Charts C and feel that the precision 

 obtained is rather better than would hav^e resulted from using formula 

 (6) for all values of / and assuming F{N, n, X, t) = \. However, 

 for suitable ranges of the variables involved, the formula resulting 

 from this procedure 



m.x)..-|:rr')(^)'(>-^r' 0.) 



would be a fairly good approximation. This is simply part of the 

 well-known binomial expansion and is far simpler to compute than 

 the more precise formulae, although by no means easy at that. 



We may draw several interesting practical conclusions, however, 

 from formula (11). For instance, we may note that as njN approaches 

 and X becomes infinite in such a way that the product (A' + \){nlN) 

 remains constant and equal to the average a, we have the familiar 

 Poisson Exponential Binomial Limit 



^ n V~" 



w{c,x) = 1 -E 



«=o ^• 

 where a = (X + 1)(«W. 



In addition we note from formula (11) that, for small values of 

 X IN, the variable N enters into the formula only in the ratio njN. 

 From this we deduce the fact, borne out by independent calculations, 

 that by means of the proper use of a proportionality factor K applied 

 directly to n and N and inversely to X jN we may extend the Charts C 

 to care for values of X IN ^ A to a very good degree of accuracy 

 and with considerable saving in space and computational labor. 



By the reciprocal relationship between A' and n as shown in exact 

 formulae (3) and (5), we obtain 



w,.)^.-f("r')(^)'(.-^,)"^'"' ^- 



which differs only in form from equation (3) of Molina's paper ^ on 

 " Footnote 1. 



