54i6 LAPLACE. 



For let us suppose that instead of (2) we have the condition 

 that x^ + x^+ ... + Xn shall lie between s and s + As. Then by the 

 theorem to which we have just referred we have 



m 1 2 n [72 + 1 



and \\\ dx^dx^,»,dxn= ; 



Hence by division we obtain 



/// 



• • • Xnn CLX., (a/X„ • • • UitL', 



jj^l^Oyj^U/tAy^ ••• ix/t*^„ f A Nn+1 W+1 



(5 + AsY^"- - s' 



• • • a*X/^ Ci/tO^ • • • Cvt^Aj 

 The limit of this expression when As is indefinitely diminished 



///■ 



is - . Then by putting for m in succession the values 1, 2, . . . r, 



we obtain the result. 



Laplace makes the following application of the result. Sup- 

 pose that an observed event must have proceeded from one of 

 n causes A, B, G, ... ; and that a tribunal has to judge from which 

 of the causes the event did proceed. 



Let each individual arrange the causes in what he considers 

 the order of probability, beginning with the least probable. Then 

 to the r^^ cause on his list we must consider that he assigns the 

 numerical value 



1 fl 1 1 



n In n — 1 n — 2 n 



-r + 1) 



The sum of all the values belonging to the same cause, accord- 

 ing to the arrangement of each member of the tribunal, must be 

 taken ; and the greatest sum will indicate in the judgment of the 

 tribunal the most probable cause. 



990. Another example is also given by Laplace, which we will 

 treat independently. Suppose there are n candidates for an office, 

 and that an elector arranges them in order of merit ; let a denote 

 the maximum merit : required the mean value of the merit of a 

 candidate whom the elector places r^^ on his list. 



