428 



Mb DE morgan, ON A QUESTION 



or, M^ = WW- loff — + 71 (I — -sr) log 



° a ' ° 1 — a 



.(7). 



= WW log ~ + n (1 - ■sr) log p— ^ 



Let a — -sr — 9, b — w + 0, where is a small fraction both of -sr 

 and 1 — •ar: then 



V = + 



: , nearly : 



and the presumption that p, the probability of the arrival of A, lies between 



■nr ± 



IS 





1 /•+/' „,, (ISrar^- 1373- + 1)^6-"' 



—7= / e~' or 7= 



y/ir-'-iy DWar(l — •5r)V7r 



In the formula of Laplace and M. Poisson, the result has 



+ 



y/'inimr (1 — w) 



instead of 



(IStjt^- 1373- + 1)m6->'' 

 6M'5r(l — ■ar) y^^ 



The latter is of a lower order than the former, and may safely be 

 rejected. By taking successively for the limits 



■zs- - 6 and u, -sr — and ■or + 6, -sr and sr + 0, 



we find that the presumption of /j lying between the several pairs of 

 limits stated in the first column is expressed by the formulae in the second. 



p lies between 



_,x/2^Il^andw 

 n 



. /27ir(l-7ir) , ^ ^/27ir(l--5r) 



- ;i V and r?- + M V ^ 



n n 



-, . /2Br(l-'5r) 



BT an 



Probability of the preceding. 



