BY E. L. PIESSE, B.SC, LL.B. 71 



Voy ^0) P> y- That is, if there is only one seat to be 

 allotted to remainders, and one party has a remainder 

 greater than half the electoral unit, that party will get the 

 seat whatever the sizes of the parties. In a similar way we 

 see that if there are two seats to be allotted to remainders, 

 and one party has a remainder less than half the electoral 

 unit, the other two parties each get one seat, whatever the 

 sizes of the parties. 



41. As the centre of gravity G, for which a = ^i = y 

 = ^y lies within AMSN , we see that when there are three 

 remainders each equal to one-third of the electoral unit, 

 and one unallotted seat, the largest party gets the seat ; 

 when there are three remainders, each equal to two-thirds 

 of the electoral unit, and two unallotted seats, the two 

 smaller parties each get a seat. 



Thus Fig. 4 is drawn for the case in which x^ = 5^ ; 

 I/q = IJ, Zq = 3J (see § 24). The ideal point is the centre 

 of gravity of the small triangle in which it lies, and also 

 the centre of gravity G of the projected triangle. As G is 

 nearer to A than to B or C, A represents the solution, 

 which is therefore (6, 1, 3). 



42. Other properties of the figure can be written down 

 from inspection, but in general to ascertain what the 

 apportionment will be it is necessary to plot the particular 

 case. Certain general results can however be seen without 

 plotting each case. Thus taking the cases in which the 

 strength of party A is 53J % (x^ — 5J, when m = 

 10), and in which the strengths of the other parties have 

 all possible values from to 46| %, we should have 

 a series of figures similar to Fig. 4. The distance of the 

 ideal point from BC will always be one-third of the perpen- 

 dicular from A to BC The distance of A from BG varies as 

 //„ and Zg change ; but the path of the ideal point can be 

 pictured as not very different from the line described by G 

 as it moves across the triangle of Fig. 4 parallel to BG. 

 This leads to the result that when the largest party is 

 53 J %, and the remainders |3 and y of the other parties 

 are each less than half the electoral unit, the largest party 

 will always get the doubtful seat, except for a very small 

 range in which [3 is just under half the electoral unit, and 

 a very small range in which y is just under half the elec- 

 toral unit. For a further discussion of this case, see § 47. 



43. In the third method of discussing the problem, 

 attention is again given to the number of votes to a mem- 

 ber, looked at in the form of the fraction of a member 

 returned by each vote. The apportionment is considered 



