S62 CONDORCET. 



To shew the truth of this equation we observe that in the 

 first p trials the following p cases may arise ; the event may- 

 happen 2^ times in succession, or it may happen p — 1 times in 

 succession and then fail, or it may happen /:> - 2 times in succes- 

 sion and then fail, , or it may fail at the first trial. The 



aggregate of the probabilities arising from all these cases is </> (r). 

 The probability from the first case is v^. The probability from 

 the second case is v^"^ ecj) (r —p) : for v^"^ e is the probability that 

 the event will happen p— \ times in succession, and then fail ; 

 and <j>(r —p) is the probability that the event will happen p 

 times in succession in the course of the remaining r—p trials. 

 In a similar way the term ?;^~V </> (r — p + 1) is accounted for ; and 

 so on. Thus the truth of equation (1) is established. 



679. The equation (1) is an equation in Finite Differences ; 

 its solution is 



* (r) = (7.2/,-+ C,y:+ C^:+ ...+ C,y;+G (2). 



Here (7^, C^, ,.. C^ are arbitrary constants ; y^ V^y "-y^ are the 

 roots of the following equation in y, 



y^ = e{v'-' +v'-'y ^v'-'f + ,.. +y'-') (8); 



and C is to be found from the equation 



0=^^ + 6(t^^-' + v^"^+... + v + l) C, 



that is (7=^^ + 6-1 G\ 



and as e = 1 — v we obtain (7= 1. 



We proceed to examine equation (3). Put 1—v for e, and 

 assume y = - : thus 



'" --^^ «. 



We shall shew that the real roots of equation (3) are nu- 

 merically less than unity, and so also . arc the moduli of the im- 

 aginary roots ; that is, we shall shew that the real roots of 



