CONDORCET. S63 



equation (4) are numerically greater than v, and so also are the 

 moduli of the imaginary roots. 



"We know that v is less than unity. Hence from (4) if z be 

 real and positive it must be greater than v. For if z be less than 



V, then _ is less than z , and a fortiori — ^ ~ is less 



V 



than -z . If s be negative in (4) we must have 1 — z^' nega- 

 tive, so that p must be even, and z numerically greater than unity, 

 and therefore numerically gi'eater than v. Thus the real roots of 

 (4) must be numerically greater than v. 

 Again, we may put (4) in the form 



v + v' + v^+ ... = z + z^-\- ...+z^ (5). 



Now suppose that z is an imaginary quantity, say 



z =zJc (cos d 4- V— 1 sin 0) ; 



then if k is not greater than v, we see by aid of the theorem 



0" = k"" (cos nd + V^ sin nO), 



that the real terms on the right-hand side of (5) will form an 

 aggregate less than the left-hand side. Thus k must be greater 

 than V. 



After what we have demonstrated respecting the values of the 

 roots of (3), it follows from (2) that when r is infinite <f> (r) = 1. 



680. We proceed to the second problem. 



Let (f) (r) now denote the probability that in r trials the event 

 will happen p times in succession before it fails p times in suc- 

 cession. 



Let ^jr (n) denote the probability that the event will happen 

 p times in succession before it fails p times in succession, supposing 

 that one trial has just been made in ivhich the event failed, and that 

 n trials remain to be made. 



Then instead of equation (1) we shall now obtain 



^(r)='yP + ?;^~'ei/r(r-;?) + v''"' eyjr (r - p + 1) -{- ... 



. . . + ve^{r (r-2) +eylr(r-V) ... (6). 

 This equation is demonstrated in the same manner as (1) w^as. 



