CONDORCET. 373 



2 



than ^. Hence by Art. 67^ the value of W/ will be unity when 



q is infinite. 



Let </) (v, ei) stand for W^, where we mean by our notation to 

 draw attention to the fact that TF/ is a symmetrical function of e 

 and ^. We have then the following result strictly true, 



(j> {v, ei) + (j) {e, vi) + <j> {i, ev) — 1. 



Now suppose q infinite. Let v be greater than e or /; then as 

 we have just shewn {v, ei) = 1, and therefore each of the other 

 functions in the above equation is zero. Thus, in fact, </> {x, yz) 

 vanishes if x be less than y 07^ z, and is equal to unity if x be 

 greater than hotli y and z. 



Next suppose v — e, and i less than v or e. By what we have 

 just seen (/, ev) vanishes ; and (/> (v, ei) = c^ {e, vi), so that each 



of them is ^ . 



Lastly, suppose that v — e — i. Then 



<^ [Vy ei) = (e, vi) = <f> [i, ev) ; 



hence each of them is ^ . 



o 



We may readily admit that wlien q is infinite W^ and W"^ 

 are each equal to TF/ ; thus the results which we have obtained 

 with respect to Problem li. of Art. 687 will also apply to Problems 

 I. and III. 



Condorcet gives these results, though not clearly. He estab- 

 lishes them for W^ without using the fundamental equation we 

 have used. He says the same values will be obtained by examining 

 the formula for TT/^. He proceeds thus on his page 10^ : Si 

 maintenant nous cherchons la valeur de TT^ nous trouverons que 

 TF* est ^gal a I'unite moins la somme des valours de TF'*, on Ton 

 auroit mis v pour e, et reciproquement v pour i, et reciproquement. 

 The words after TF'* are not intelligible; but it would seem that 

 Condorcet has in view such a fundamental equation as that we 

 have used, put in the form 



(/) {v, ei) = 1 — <^ (e, vi) — (j> {i, ev). 



But such an equation will not be true except on the assumption 



