306 



Transactions. 



Table VI (First Trial Table). 



D and E, having trial totals not greater than the average, are rejected, and 

 the columns and lines for D and E are deleted. 



The second trial table, which for clearness is here written out again, 

 appears as follows : — 



Table VII (Second Trial Table). 



A's and B's trial totals being not greater than the average of the trial 

 totals (400), they are rejected, and C is declared elected. 



The soundness of this method can be proved independently of Nanson's 

 method ; for the successful candidate is, generally, the candidate who would 

 be preferred to any other single candidate if they were the only two candi- 

 dates : hence if n, which must be either of the form 2N or of the form 

 2N + 1, be the total number of voting-papers, the successful candidate 

 must as against any other candidate obtain at least N + 1 votes, and his 

 trial total must not be less than (m — 1) (N + 1), where m is the number of 

 candidates. These total preferences occur in pairs, and the sum of each pair 

 is n — e.g., in Table VI the total preferences of A over B is 210, and of B 



over A 190, and their sum is 400. The number of pairs is m \ n ~ ', therefore 



the average of all the trial totals is \nm (m — 1) -r m = \n (m — 1), which 

 is either of the form [m — 1) N, or of the form (m — 1) (N + |). Any 

 trial total not greater than this is less than (m — 1) (N + 1). A candidate 

 with a trial total not greater than the average of all the trial totals cannot, 

 therefore, be elected, and may be thrown out at any stage. 



I have said that generally the successful candidate is the candidate who 

 would be preferred to any other single candidate if they were the only two 

 candidates ; but there are two other cases — namely, the case of equality, 

 and what Nanson calls the inconsistent case. In each of these cases the 

 same rule as that given above applies. 



