S54* CONDOECET. 



becomes when q^ is changed into $' -H 1, that is let ^ (g + 1) denote 

 the probability that there will be a majority in favour of a correct 

 decision when the question is submitted to 2^' + 8 voters. There- 

 fore 



<^ (^ + 1) = ^,^^- + (2^ + 3) v^-'^ e + (?i±|I|i±21 ^...i ,. 



+ ...+ 



2^ + 3 



(7 + 2 q + 1 



v'"-' e'^\ 



' Since v-\-e = l we have 



^ (2) = (« + «)" <!> fe). 



Thus ^(j + l)_,/,(j) = ^(2 + l)_(„ + e)»^(j). 



Now (f> (q + 1) consists of certain terms in the expansion of 

 (v + ey^^^, and cp (q) consists of certain terms in the expansion of 

 {v + ey^'^^ ; so we may anticipate that in the development of 

 (j> (q+l) — (v + ey (j) (q) very few terms will remain uncancelled. 

 In fact it will be easily found that 



I 2^ + 1 \2q + l 



g + 1 [I |g + l |g 



2q + l 



l+i [9 



Hence we deduce 



^ (^) = y + (v -e) jve + ji;V+ J^^'^'+ 7V3 ^*^* 



... + r==f^v (2). 



664<. The result given in equation (2) is the transformation 

 to which we alluded. We may observe that throughout the first 

 part of his Essay, Condorcet repeatedly uses the method of trans- 

 formation just exemplified, and it also appears elsewhere in the 

 Essay ; it is in fact the chief mathematical instrument which 

 he employs. 



It will be observed that we assumed v + e = l in order to 

 obtain equation (2). We may however obtain a result analogous 



