BOO CONDORCET. 



be a valid and correct decision, and -^/r (n) the probability that 



there will be a valid and incorrect decision ; let v and e have the 



same meaning as in Art. 662. Then, when n is infinite, if v is 



2 , , 2 



greater than ^ we have ^ (n) =1, if v is less than ^ we have 



2 

 (w) = ; and similarly if e is greater than ^ > that is if v is 



o 



1 . . 2 . 



less than ^ , we have -^ (n) = 1, and if e is less than -^ , that is 

 o o 



if t; is greater than ^ , we have ylr (n) ==0. 



o 



We shall not stop to give Condorcet's own demonstrations of 



these results ; it will be sufficient to indicate how they may be 



derived from Bernoulli s Theorem; see Art. 123. We know from 



this theorem that when n is very large, the terms which are in 



the neighbourhood of the greatest term of the expansion of 



{v-\-eY overbalance the rest of the terms. Now </> {n) consists of 



the first third of all the terms of (v + e)", and thus if v is greater 



2 

 than - the greatest term is included within <j> (n), and therefore 



(f> (n) =1 ultimately. 



2 

 The same considerations shew that when v = -^, we have 



1 . ^ 



^(n) = ^ ultimately. 



675. Condorcet's seventh and eighth Hypotheses are thus 

 described by himself, on his page xxxiii : 



La septieme hypothese est celle ou I'on renvoie la decision a un autre 

 temps, si la pluralite exigee n'a pas lieu. 



Dans la huitieme hy]^)othese, on suppose que si I'assemblee n'a pas 

 rendu sa premiere decision a la pluralite exigee, on prend une seconde 

 fois les avis, et ainsi de suite, jusqu^a ce que Ton obtienne cette pluralite. 



These two Hypotheses give rise to very brief discussions in the 



Essay. 



676. The ninth Hypothesis relates to the decisions formed 

 by various systems of combined tribunals. Condorcet commences 

 it thus on his page 57 : 



