872 CONDORCET. 



689. Condorcet does not discuss these problems with much 

 detail. He gives some general considerations with the view of 

 shewing how what he denotes by W^^^ may be derived from TV^; 

 but he does not definitely work out his suggestions. 



We will here establish some results w^hich hold when the 

 number of voters is infinite. 



We wdll first shew that when q is infinite W/ is equal to unity, 

 provided that v is greater than either e or i Suppose {v + e+iY^'^^ 

 expanded in the form 



{v + 6)^^^ + {6q + l){v + er i + ^t^-^ (^ ^. ey 



16^41 

 ' 4^ + 1 zg' ^ ' 



\Gq-l '2 



^^ + 



Now take the last term which we have here explicitly given, 

 and pick out from it the part which it contributes to W^. 



We have {v + e)*^"^^ = {v + eY'"-' -f- + -— , 



Expand \ 1 \ as far as the term which involves 



f V \^^^^ f V e \ 



, and denote the sum by / , J . Then finally 



the part which we have to pick out is 



\6q±l 





+ ey 



(V 6 \ 

 , ) is equal to 



unity when q is infinite, as we have already shewn ; see Art. 660. 



Hence we see that when q is infinite the value of W/ is the 

 limit of 



{v + e)^^^ + {6q + l){v + 6)^ I + ^^^ ^ ^J ^^ {v + e)*^-^ i' + 



16(7 + 1 



Now we are at liberty to suppose that { is not greater than e, 

 and then i?4-6 is greater than 2i; so that v-\-e must be greater 



