302 LAGEANGE. 



error equal to — 1 is made ; it is required to find the probability 

 that in taking the mean of n observations, the result shall be 

 exact. 



In the expansion of {« + 5 (ic + cc~^)}" according to powers of x, 

 find the coefficient of the term independent of x\ divide this 

 coefficient by {a + 2^)" which is the whole number of cases that 

 can occur ; we thus obtain the required probability. 



Lagrange exhibits his usual skill in the management of the 

 algebraical expansions. It is found that the probability diminishes 

 as n increases. 



558. We may notice two points of interest in the course of 

 Lagrange's discussion of this problem. Lagrange arrives indirectly 

 at the following relation 



„ [n (n — 1)1 ^ {n(n — 1) (n — 



2.3 J"^*" 



^ 1.3.5... (2n~l) , 

 1.2.3...71 



and he says it is the more remarkable because it does not seem 

 easy to demonstrate it a i^riori. 



The result is easily obtained by equating the coefficients of the 

 term independent of x in the equivalent expressions 



(1 + ^rfl + iy, and^li^ 



^2n 

 XI ' X'' 



This simple method seems to have escaped Lagrange's notice. 



Suppose we expand ■ in powers of z ; let the 



V 1 — 2as — cz^ 



result be denoted by 



1 + A^z + A/ + A/ -f ... ; 



Lagrange gives as a known result a simple relation which exists 

 between every three consecutive coefficients ; namely 



. _2n — l . n — 1 J 



