364 CONDORCET. 



We have now to shew the connexion between the functions 

 (j) and yjr; it is determined by the following relation ; 



'>lr(n)==(l) (n) - e^"^ [cf> {n-p +l)-ef (n-p)} (7). 



To shew the truth of this relation we observe that yjr (n) is 

 less than <f> (n) for the following reason, and for that alone. If the 

 one failure had not taken place there might be ^ — 1 failures in 

 succession, and there would still remain some chance of the 

 happening of the event p times in succession before its failing 

 p times in succession ; since the one failure has taken place this 

 chance is lost. The corresponding probability is 



e^-' {(f> (n -p + 1)- ef {n -p)}. 



The meaning of the factor e^~^ is obvious, so that we need only 

 explain the meaning of the other factor. And it will be seen 

 that (j) (n — p -h 1) — eyjr {n — p)) expresses the probability of the 

 desired result in the n—p + 1 trials which remain to be made; 

 for here the rejected part eyjr{n—p) is that part which would 

 coexist with failure in the first of these remaininof trials, which 

 part would of course not be available when p—1 failures had 

 already taken place. 



Thus we may consider that (7) is established. 



In (6) change r into r —p ; therefore 



^ (r-p) = v^ + v'^'^ef {r-2p) + if-' e^^r [r-2p-\-V) + ... 



. . . + ve-^ (r —p — 2) + ei/r (r —p — 1) (8). 



Now multiply (8) by e^ and subtract the result from (6), ob- 

 serving that by (7) we have 



i/r in) — e^'yjrin —p) = ^ (n) — e^"* <^ {n —p + 1) ; 



thus we obtain 



<f> ir) - e^ ^ (r -p) =v^ - eV 



+ v""'' e {(p {r -p) - e^-' <^ (r - 2p + 1)} 

 + v''-'e{<\>{r-p-\-l) -e^"'^ (r-2^+2)} 

 + ... ' 



j^e[^(r-r)-e"<t>{^-p)] (9). 



681. The equation in Finite Differences which we have just 



