102 BELL SYSTEM TECHNICAL JOURNAL 



by the existing tiibles for the probability integral. However, this 

 difficulty is encountered only when P, or (1 — P), is small, in which 

 case 7' is large and the integral 



I 



T 



e-^-dt 



may be readily e^'aluated by computing the first few terms of the 

 series 



Ie-T'I2T-{^J_\ - Tx-- + (l-3)rr^ - (i-3.5)rx-« . • •]- 



where, as above, 7\ = T-sl. 



When P is very small, the difference c — a = c — np \s relatively 

 large compared to a, and for this latter case recourse may be had to 

 the approximate formula published by the writer in the American 

 Mathematical Monthly for June, 1913. 



Derivation of the Approximate Formula 

 Following Laplace closely, let us set 



(6) y{x) = Ye-'\ 



where I' = 3'(.To) is the maximum value of y(.v). Then 



(7) f-^*- = ^-£-'*(l)'"' 



the upper limit T being given by the equation 



(8) y{p) = y(xo)e-'^\ 

 Assuming dx/dt expanded in powers of / so that 



(9) dx/dt = J^Ds+it' 



and setting R^ = A4-1/-C1, equation (7) reduces to 

 f ydx = YDiJ^Rs f t'e-^'dt. 



Jo s=0 •' — 00 



Our fimdamental equation (1) may now be written 



Y..Rs f t'e-^'dt 



s=0 J-x 



(10) Z(l)p'(^ -py-^ = 



'-' ^ ' y.R, fe-"dt 



HR.r 



S = J — Q 



