LAPLACE. 547 



Let ^1, t^, ... tn denote the merits of the candidates, beginning 

 with the most meritorious. The problem differs from that just 

 discussed, because there is now no condition corresponding to the 

 sum of the ordinates being given ; the elector may ascribe any 

 merits to the candidates, consistent with the conditions that the 

 merits are in order, none being greater than that which imme- 

 diately precedes it, and no merit being greater than a. 



The mean value of the merit of the r*^ candidate will be 



///•• 



vv CtL^ (^^n . • . Cvt', 



n 



• • • etc, ^^o • • • ^^n 



The integrations are to be taken subject to the following con- 

 ditions : the variables are to be all positive, a variable ^^ is never 

 to be greater than the preceding variable t^^, and no variable is to 

 be greater than a. Laplace's account of the conditions is not in- 

 telligible ; and he states the result of the integration without 

 explaining how it is obtained. We may obtain it thus. 



X Ut t,i = X^, fn_^ = ^n + ^n-i? ^n-2 ~ ^n-i + ^n-2> '" l tneU the 



above expression for the mean value becomes 



III • • • \p^n ~r" ^n—i i • • • "i~ "^r) ^«^i CiOS^ ... CIX^ 



with the condition that all the variables must be positive, and 

 that iCj + cCg + ••• + ^n i^^st not be greater than a. Then we may 

 shew in the manner of the preceding Ai'ticle that the result is 



{n — r-VV) a 



Laplace suggests, in accordance with this result, that each 

 elector should ascribe the number n to the candidate whom he 

 thinks the best, the number n—\ to the candidate whom he 

 thinks the next, and so on. Then the candidate should be 

 elected who has the greatest sum of numbers. Laplace says, 



35—2 



