220 Methods of Election. 



of a, ft, y, so that although for given values of a, b, c we 

 cannot bring about any result we please, still we can choose 

 ■a, ft, y so as to bring about a result different from the true 

 one. This, of course, is to be done by choosing a, ft, y, so 

 that the best candidate is thrown out at the first scrutiny. 

 We have already seen that this is possible. 



It is clear that no one of the quantities ft -\- y, 7 + a, a + /3 

 can be negative. For we have ft + y = BC + CB, and 

 BC, CB can neither of them be negative. Again, ft + y = 

 N — a ; thus a cannot be greater than N. So also ft, y can 

 neither of them exceed N. Since ft + y cannot be negative, 

 ft and y cannot both be negative ; thus one only of the three 

 a, ft, y can be negative. If a be negative it is clear that the 

 numerical value cannot exceed N, for a + ft cannot be nega- 

 tive, and ft cannot exceed N. So for ft and y. Thus no 

 one of the three a, ft, y can numerically exceed N, and one 

 at most can be negative. 



The limits between which a, b, c must lie are at once 

 found from the consideration that AB, AC, &c, must none 

 of them be negative. Thus a + y, ft — a can neither of 

 them be negative ; thus a cannot be less than — y nor 

 greater than ft. Hence, a fortiori, no one of the three 

 a, b, c, can be numerically greater than N. This last result 

 is obvious from the fact that no one of the numbers in the 

 columns headed " Condorcet " can be negative. 



Formal demonstrations will now be given of a few results. 



(i.) If any candidate have less than N votes on the 

 double vote method, he ought not to be elected. 



This has already been seen, but the following proof is 

 given. Suppose A has less than N votes; then a must be 

 negative, and therefore c must be negative and b positive. 

 Thus A is worse than B, and also worse than C. 



(ii.) Even if every elector put A in the first or second 

 place it does not follow that A ought to be elected. 



For if A has no third places we must have BC = and 

 CB = 0, thus a = ft = — y. Suppose ft positive and there- 

 fore y negative. Then by preceding case C ought to go out 

 and A or B ought to win as c is positive or negative. Now 

 c may be negative so that B may win; for the only conditions 

 with reference to c are that c must be greater than — ft and 

 less than a, and as ft is positive it is clear that c may be 

 negative. 



(iii.) It is impossible to arrive at the true result by 

 merely counting the number of first places, the number of 



