LAPLACE. 551 



2 J II -Ell 



Here y = ——-!-— -e 2""» and the differential coefficients of v 



with respect to r will introduce the factor -—- — , and its powers ; 



and ^-— is of the order —j- at most, so that when multiplied by 



the constant factor in y we obtain a term of the oraer — . Thus 

 as far as we need proceed, 



where both the symbols S and I are supposed to indicate opera- 



1 



tions commencing with «• = 0, and ^ Y denotes the gi'eatest term 



1 



of the binomial expansion, that is the value oi ^y when r = 0. 



The expression ^y denotes as usual the sum of the values of y up 

 to that corresponding to r — 1 ; adding the value of y correspond- 

 ing to r we obtain 



jydr-v^y + -^Y\ 



subtract the greatest term of the binomial, and thus we have 



I 



ydr-^^y. 



/y kI 11, 



Put T = ,.^ . ; thus we obtain finally 



2 f ... 7, V/^ 



-2 



This expression therefore is the apj^roximate value of the sum of 

 2r + 1 terms of the expansion of {p + qY, these terms including 

 the greatest term as their middle term. In the theory of proba- 

 bility the expression gives the probability that the number of 

 times in which the event will happen in //, trials will lie between 

 m — 'T and m + r, both inclusive, that is between 



T \/(2mn) T , , r\/(2mn) 

 ^^^.^^_V^__and/x^ + ^4--^^; 



