218 



Methods of Election. 



" B is better than C," which is affirmed by a majority of 2a r 

 which is smaller than the former majority. 



There is, however, one point in favour of the modified 

 method. The first scrutiny will at once give us the values 

 of the three differences b — c, c — a, a — b. From these, of 

 course, we cannot find a, b, c. In the modified method, 

 however, a second scrutiny is always necessary, and this will 

 at once give us the value of one of the three a, b, c. Having 

 already found the three differences, we can at once find each 

 of the quantities a, b, c, and hence we can ascertain if the 

 result is demonstrably correct. Thus if the modified method 

 be used, we can always ascertain, by a simple calculation, 

 whether the result is perfectly satisfactory or not. The 

 same remark applies to the proposed method in those cases 

 in which two scrutinies are necessary. 



Algebeaic Analysis. 



Before leaving the case in which there are three candidates 

 only, it may be of interest to give a short algebraical analysis 

 of the question. As before, let the compound symbol AB 

 stand for the number of electors who put A first and B 

 second, and similarly for other cases. Let us suppose, as is 

 clearly possible, that six quantities, a, b, c, a, /?, y, are found 

 from the following equations : 



AB=/3 + c BO= 7 +« CA = a+b 

 AC = y-b BA-a-c CB = /3-a 



Also let us suppose that 2N denotes the whole number of 

 electors, which is clearly equal to 2 (a + y8+y), then the states 

 of the poll on the different modes of election which have 

 been discussed are as shown in the following table : — 



Analysis of Votes. 



Single Vote. 



Double. 



Borda. 



Condorcet. 



jAB = /3 + *) 

 A \ AC=y-b \ 



„ \BC=y+a{ 

 D (BA=a-c j 



n j CA=a+b j 



(3 + y— b + e 

 y+a—c+ a 

 a + (3-a+b 



N+a 



N + /3 

 N + 7 



2^-b+c 

 2N-c + a 

 2^-a+b 



N + a 

 TS-a 



N-5 



* 



N + & 



N + c 

 N-c 



2N = 2(a + /3 + 7 ) 



2N 



4N 



6N 



2N 



2N 



2N 



