and the Elimination of Chance. 311 



have the usual degree of approximation,* in which the lowest 



terms rejected are of the order -. This condition may be 



thus translated into the usual postulate, that the excess 

 should not exceed the order <s/pu. Expanding k in terms of 



p, where p = ^~n n, we have 



# +A (?r } 



Now, when p is of the order \/1t, since terms of the order - 



r V 2 fi 



may be neglected, we have k of the order unity. But, when 



p exceeds the order — -=, since k increases with p, k exceeds 



the order unity. Hence the ordinary limitation is from this 

 point of view perceived to be both sufficient and necessary. 



It is more important for our purpose to show that the 

 ordinary formulae of approximation hold good far beyond the 

 limit ordinarily prescribed, not of course with what may be 

 called the regulation degree of precision (where the smallest 



order neglected is -), but with sufficient precision for prac- 



/* _ 



tical purposes. Suppose p is of a higher order than \Zpu, e.g. fd; 

 then k is of the order /xi The series \' + \"k + &G. is still 

 convergent. It is still possible to simplify our formula by 



k 2 

 expanding k and neglecting terms of the order — r And the 



A 6 

 further simplification which is necessary to reduce Poisson's 

 first formula to the more usual one given, e. g. by Mr. Tod- 

 hunter, is still allowable. We have thus for a first approxi- 

 mation to the sought probability, 



dt-\ = e~ T2 ; where r= — ^-j= 



x/27TfA, /p. 



(the Poissonian X u for the case of symmetry). 



For a second approximation we have the more rugged 

 formula, from which Poisson reasons downwards, 



* Cf. Todhunter, he. cit. 



t This expression for the tail of the binomial locus is deducible from 

 Mr. Todhunter's formula for its body (History, art. 993). 

 % Ibid. art. 79. 



tj_r 



Vir) T 



e~ i2 



