Methods of Election. 219 



In the first column is set out an analysis of the votes. 

 In the second is the result of the poll on the single vote 

 method. For instance, in the first line we have the quantity 

 /?-{-y-6 + c, which is the sum of AB and AC, i.e., it denotes 

 the number of electors who put A first. In the third 

 column is the result of the poll on the double vote system, 

 in which each elector has two votes.' For instance, in the 

 first line we have N + a, or what is the same, 2a + j3 + y, and 

 this is equal to AB + AC + BA + CA, i.e., it denotes the 

 number of electors who put A first or second. In the 

 fourth column is the result of the poll on Borda's method. 

 For instance, in the first line we have 2N — b + c, and this is 

 equal to 2 AB + 2 AC + B A + CA, as it ought to be. It is 

 also seen at once that 2N — b + c is the sum of the two 

 numbers in the first line in the second and third columns. 

 This shows the truth of what was stated above, viz., that 

 the poll on Borda's method is the aggregate of the polls on 

 the single and double vote systems. In the fifth, sixth, and 

 seventh columns, under the heading Condorcet, are set 

 down the states of the poll on the supposition that each of 

 the candidates, A, B, C, is excluded in turn. Thus, if A be 

 supposed excluded for a moment, we have N + <x votes for 

 B in preference to C, and consequently N — a for C in pre- 

 ference to B. For N+a is equal to AB + BC + BA, as it 

 ought to be. Thus it is clear that 2a is the majority for 

 B as against C, so that the letters a, b, c, have the same 

 meaning as in the previous part of this paper. It is clear 

 too, as has been proved before, that the number in any row 

 in the column headed Borda, is the sum of the two numbers 

 in the same row in the columns headed Condorcet. 



The result of the method of election proposed in this 

 paper depends solely upon the numbers a, b, c. The same is 

 true of the method of Borda. On the other hand, the result 

 of the double vote method depends solely on the values of 

 <*-> P» y. Consequently, whatever be the result of the pro- 

 posed method or of Borda's method we can clearly construct 

 cases in which the result of the double vote method shall be 

 what we please. The same is true of the single vote 

 method ; for although the result of the single vote method 

 depends upon a, b, c as well as upon a, /3, y, it is easy to 

 see that we can choose a, fB, y so as to eliminate the eftect 

 of the quantities a, b, c, whatever may be the values of the 

 latter. The results of the Venetian method and of Ware's 

 method depend on the values of a, b, c as well as upon those 



