Methods of Election. 227 



be given to B,, B 2 , &c, C b &c. If A x be greater than N, there 

 is an absolute majority for A, and we may at once elect 

 him. If A n be greater than N, there is an absolute majority 

 against A, and we may at once exclude him. If neither of 

 these results hold good for any candidate, we must use the 

 proposed method in its general form. Now A's score on 

 that method is 

 (n — \)A X + (n — 2)A S + . . ' . + (n — r)A r + . . . +A M _ V 



Thus to find A's score we must find A 2 , A 3 . . . A a _,. Now 

 to find these it is not necessary to count all the votes for A. 

 For we have 



A x + A 2 + A 3 + . . . + A„ = 2N, 



and A b A„ having been already found, we see that it is 

 sufficient to calculate any n — 3 of the n — 2 quantities, 

 A 2 , A 3 . . . A w _ 1 , and the remaining one can then be found 

 from the above equation. 



It would, however, in practice be better to calculate each 

 of the n quantities, A 1? A 2 . . . A n , and then to use the above 

 equation as a test of the accuracy of the counting of the 

 votes. Similar remarks apply to the numbers Bi, B* . . . B„, 



c 1; c 2 . . . c„, &c. 



We have also n equations of the former 

 A r + B r + C, + • • • = 2N 

 where r may have any one of the values 1, 2, 3 . . . n. This 

 gives us n independent tests of the accuracy of the 

 enumeration of the votes. In fact, if we arrange the 

 n 2 quantities, A h A 2 . . . A m B b &c., in the form of a square 

 array 



A 1? A 2 , A 8 , &c. 



B b B 2J B„ &c. 



C„ C 2 , C 8 , &c. 



ifec, &c, &c. 



the sum of every row and of every column ought to be 2N, 

 so that we have altogether 2n — 1 independent tests of the 

 accuracy of the enumeration of the votes. 



The proposed method is not so laborious as might appear 

 at first sight. The number of scrutinies will not usually be 

 large ; for Ave may reasonably expect to halve the number 

 of candidates at each scrutiny. At each scrutiny we reject 

 all who are not above the average. Now in the lono- run 

 we may expect to find as many below as above the average 

 on a poll. Thus, if there be eight candidates we should 



K 2 



