B6 QUOTA IN PROPORTIONAL REPRESENTATION, 



The numbers x^ y, z are integral; and the positive 

 integral solutions of (4) are represented by the points of 

 intersection on ABC of planes drawn parallel to the 

 coordinate planes, and at distance 0, 1, 2 . . .m from 

 them. The solutions are therefore represented by the 

 nodes of the equilateral triangular lattice shown in Fig. 

 3; and their numerical values are proportional to the dis- 

 tances of the nodes from the sides of ABH 



Fig. 3. 



28. The ideal solution Xq, y^^ Zq, is represented 

 Ijy a point I (called the ideal point) in this triangle. If 

 the rule-of -three method can be used without allotment of 

 seats to remainders, the point / is a node. If allotment 

 of seats to remainders is necessary, / is not a node, and 

 we have to determine which of the neighbouring nodes 

 gives the solution. 



29. Let the triangle ABC be drawn so that the perpen- 

 diculars from A, B, C to the opposite sides are each m. 

 The X, y, z of any point are then equal to the distances 

 of the point from the sides of ABC, that is, they 

 are the trilinear coordinates of the point. For 

 instance, the point / in Fig. 3 represents the ideal 

 solution in a case in which the strength of party A is 

 exactly five times the electoral unit, the strength of party 

 B, between one and two times the electoral unit; and the 

 strength of party C between three and four times the elec- 

 toral unit. Thus x^ = X = perpendicular from QR to 

 BG ; 2/o = perpendicular from I to AB -, Y = perpendicu 

 lar from PQ to AB ; (5 = perpendicular from / to PQ ; 



Zq = perpendicular from 7 to ^C; ^ = perpendicular 

 from PR to AG ; y = perpendicular from I to PR. 



