CONDORCET. 859 



supposing the numbers for and against not to be known ; that is 



. / N . / \ must be large. 



5. That a vaHd decision which has been given is correct, 

 supposing the numbers for and against to be known ; that is 



-jjp^^ :j^^-^ must be large, even when m and 7i are such as to 



give it the least value of which it is susceptible. 



These appear to be what Condorcet means by the principal 

 conditions, and which, in his usual fluctuating manner, he calls 

 in various places Jive conditions, four conditions, and tivo con- 

 ditions. See his pages xviii, xxxi, LXix. 



672. Before leaving Condorcet's second Hypothesis we will 

 make one remark. On his page 17 he requires the following 

 result. 



{l + ^(l_4^)p^(l_4^) ^' 1 -^ • 1.2 



I w + 2r — 1 



• • • T ', ; ; T ^ "!"••• 



7' \7i + r— I 



On his page 18 he gives two ingenious methods by which the 

 result may be obtained indirectly. It may however be obtained 

 directly in various ways. For example, take a formula which may 

 be established by the Differential Calculus for the expansion of 

 (1 + \/(l — 4^)}""^ ii^ powers of s, and differentiate with respect 

 to z, and put n — 2 for 771. 



673. Condorcet's third Hjrpothesis is similar to his second ; 

 the only difference is that he here supposes 2q voters, and that 

 a plurahty of 2q is required for a valid decision. 



674<. In his fourth, fifth, and sixth H^^otheses Condorcet 

 supposes that a plurality is required which is proportional, or 

 nearly so, to the whole number of voters. We will state the 

 results obtained in one case. Suppose we require that at least 

 two-thirds of the whole number of voters shall concur in order 

 that the decision may be valid. Let n represent the whole num- 

 ber of voters ; let ^ (n) represent the probability that there will 



