70 



QUOTA IN PROPORTIONAL REPRESENTATION, 



of the projected triangle ABC ', these will be found to bfr 

 given by 



vv 



X, 



VI' 



IIV 



^,2 ^2 -, 



• 02 



^.2 



2 ■"" -^ ~0 ~9 I 



(10) 



.2 -V 



^-l 9 9 













Writing cos A, cos B, cos C in terras of a, h, c, and so 

 of m, ^0, 2/oj ^0, we find that each is positive, and not 

 greater than 1. The triangle is therefore acute; and (10) 

 show that the greatest side is the base from which the 

 co-ordinate of the greatest party is measured. One pos- 

 sible shape of the triangle is shown in Fig. 4. 



Fig. 4. 



Other properties of the triangle are that the circum- 

 centre S is within the triangle ; and that if x^ is the great- 

 est of a?(j, 2/o, Zq, the centre of gravity G is within the 

 triangle MNS. 



It is to be noted that areal coordinates project unaltered, 

 and that any line from a vertex to the opposite side is 

 divided by a line parallel to the base in the same ratio after 

 projection as before. 



The region AMSN is the portion of the triangle in which 

 each point is nearer to A than to 5 or C; so with BLSN 

 and GLSM, 



All these properties are true of the projection of the 

 small triangle in which I lies; let us now consider Fig. 4 

 as representing the projection of the small triangle. 



40. If the small triangle is similarly situated to 

 the large triangle, and a, or ^^ — X, is greater than J, the 

 ideal point will be in the region ANM, and so 

 will be nearer to A than to B or (7, whatever the values of 



