CONDORCET. 407 



soit pour juger dii plus on clii moins de coiifiance que meritent les 

 differens systemes imagines pour concilier entr'elles des ^poques qui 

 paroissent se contredire. 



Condorcet names Freret as having opposed this apphcation of 

 the Theory of Probabihty, and Yoltaire as having supported it ; but 

 he gives no references. 



747. According to some historians the whole duration of the 

 reigns of the seven kings of Rome was 257 years. Condorcet pro- 

 poses to examine the credibility of this statement. He assumes 

 that in an elective monarchy we may suppose that a king at the 

 date of his election will be between 30 years old and 60 years old. 

 He adopts De Moivre's hypothesis respecting human mortality ; 

 this hypothesis, as Condorcet uses it, amounts to assuming that 

 the number of people at any e230ch who are y years old is 

 h (90 — ?/), where h is some constant, and that of these Iz die every 

 year. 



Let n denote the greatest number of years which the youngest 

 elected king can live, m the greatest number of years which the 

 oldest elected king can live ; then the probability that a single 

 reign will last just r years is the coefficient of ^ in the expan- 

 sion of 



ill - m -\-X)x(X-x)- .t"^^ + rr"^" 



A few words will be necessary to shew how this formula can be 

 verified. It follows from our hypothesis that the number of per- 

 sons from whom the king must be elected is 



h [n + (?i - 1) + (?i - 2) + . . . + m], 

 that is Iz — ^— [n — m 4- 1). And if r be less than m + 1 the num- 

 ber of persons who die in the r*'^ year will be I: {n — m + 1) ; if r be 

 between m 4- 1 and n+1, both inclusive, the number who die in 

 the r^^ year will be k {n — r + 1) ; if r be greater than n + 1 the 

 number who die in the r"' year will be zero. Now the coefficient 

 of ic*" in the expansion of 



1-x (1 - -^j' 



