and the Elimination of Chance. 313 



is the method employed by the present writer in a former 

 number of this Journal *, not without reference to the kindred 

 work of Mr. Morgan Crofton. The essential feature of this 

 method is a partial differential equation which must be ful- 

 filled by the general expression for the probability of diver- 

 gence x. In the case of a binomial let u sx denote that pro- 

 bability, where s, an integer, = ^fju, and consider the formation 

 of u s+1}Z . We have 



Whence 



an equation in finite differences which may under certain 

 provisos be reduced to the differential equation 



du_l d%u m 



ds 4 da? ' 

 an appropriate solution of which is found to be 



1 _f^ 



v- 



s 



,,«•<?•(!)• 



In order that this solution should satisfy the equation infinite 

 differences with a precision which retains quantities above 



the order -, it will be found necessarv that a 2 should not 



exceed ~. But in order that some degree of precision should 



be attained, it is sufficient that x be of an inferior order to 



Thus from this point of view also it is seen that 



the received formula is fairly accurate beyond the regulation 

 limit. 



And further, when it ceases to be accurate, it still remains 

 safe : inasmuch as it affords a value superior to that of the 

 real probability of mere accident; and therefore underrates 

 the evidence in favour of the demonstrandum, the existence of 

 a disturbing cause other than chance. To show this it is 

 almost sufficient to compare the extreme term of the binomial 

 locus with the corresponding ordinate of the probability curve. 

 The Napierian logarithm of the former is — /ul log e 2 ; that of the 



latter — ~— ^ log k — 9 l°g 7r - The latter is therefore greater 



than the former, and the curve has left the discontinuous locus 

 underneath it before it reaches the extreme region of the locus. 



* Oct. 1883. 



