Methods of Election. 221 



second places, and the number of third places for each 

 candidate. 



This result seems obvious enough after what has been 

 given. It may, however, be formally proved as follows. 



Let A 1; A 2 , A 3 denote the numbers of first, second, and 

 third places respectively for A, and let corresponding 

 meanings be given to B 1? &c., C b &c. Then we have 



A 1 = /?-f-y — b + c 



A 2 = 2a + b — c 



A 3 = p + 7 

 with corresponding equations for B's and C's. We see at 

 once from these equations that it is impossible to find a, b, c 

 even if A 1? A 2 , A 3 , B ly &c, be all given. We can, however, 

 find a, ft, y and the three differences b — c, c — a, a — b, 

 viz., the results are 



a = N - A 3 , p = N - B 3 , y = N - C 3 

 b — c = A 3 — A D c — a = B 3 — B 1? a — b =. C 3 — Cj, 

 where 2N = A x + B x + O x = A 3 + B 3 + C 3 . . . . (i) 

 thus any five of the quantities A 19 B 1; C 1? A 3 , B 3 , 3 , may be 

 chosen at pleasure ; the sixth and N are then determined by 

 the conditions (i) and A 2 , B 2 , C 2 are then given by the 

 equations 



A 2 = 2N — A x — A 3 , &c. 



(iv.) If there be a demonstrably correct result, say A 

 better than B and B better than C, so that c, a, are positive 

 and b negative, then if Ware's method be wrong, Venetian 

 method is right, and if Venetian method be wrong, Ware's 

 method is right. 



For if Ware be wrong A must be lowest on the single vote 

 method, and therefore we must have 



a + fi — a + b >/? + y — b + c 



or a > y | a -f c — 2b 

 i.e., a fortiori a > y because a, c are positive and b 

 negative. Thus A cannot be lowest on double vote method, 

 so that A will win on the Venetian method. Again, if 

 Venetian be wrong, A must be lowest on double vote method, 

 and therefore we must have y > a and therefore /? + y — b + 

 c > a + /3 — a -{- b because a, c are positive and b negative. 

 Thus A cannot be lowest on single vote method, so that A 

 will win on Ware's method. 



(v.) If we agree to accept the proposed method as correct 

 in all cases, then the conclusions of the last proposition will 

 be true in all cases. 



