CONDORCET. 365 



obtained may be solved in the ordinary way ; we shall not how- 

 ever proceed with it. 



One case of interest may be noticed. Suppose r infinite ; then 

 fj) (^r—p), (/> (r — 2/? + 1), ... will all be equal. Thus we can obtain 

 the probability that the event will happen p times in succession 

 before it fails p times in succession in an indefinite number of 

 trials. Let F denote this probability ; then we have from (9), 



-e^F(t;^-'+^;^-'+... + v + l). 

 Hence after reduction we obtain 



F= ^^-^g-o (10) 



682. The problems which we have thus solved are solved by 

 Laplace, Theorie...des Proh. pages 2^7 — 251. In the solution 

 we have given we have followed Condorcet's guidance, with some 

 deviations however which we will now indicate ; our remarks will 

 serve as additional evidence of the obscurity which we attribute 

 to Condorcet. 



Our original equation (1) is given by Condorcet ; his demon- 

 stration consists merely in pointing out the following identity ; 



(v + ey = 'if{v +e)'^ + v'-'e {v + e)'^ + v'^e {v + e)"^^' + ... 

 ...+v''e{v + ey-^ + ve {v + e)'""' + e (t; + e)^-\ 



He arrives at an equation which coincides with (4). He shews 

 that the real roots must be numerically greater than v ; but wdth 

 respect to the imaginary roots he infers that the moduli cannot 

 be greater than unity, because if they were </> (r) would be infinite 

 when T is infinite. 



We may add that Condorcet shews that (4) has no root which 



is a simple imaginary quantity, that is of the form a v — 1. 



If in our equation (7) we substitute successively for ^/r in t^rms 

 of <^ we obtain 



-i/r (r) = </) (r) - e^"' {0 [r -^ + 1) - ec/) [r - p)] 



- i'-^ [^ (r - 2^ + 1) - ec/) (r - 2/?)} 



- i^-^ {cf> (r -Sp + l)-e<l> {r - Sp)} 



