BY E. L. PIESSE, B.SC, LL.B. 67 



/ may be either in a small triangle (such as FQR) simi- 

 larly situated to ABC, or in a small triangle (such as 

 P'Q'R) not similarly situated to ABC. In the first case, 

 if we move from I to P we do not alter J or ^, but we 

 increase X to X + 1 ; in the second case, if we move from 

 I to F' we do not alter X, but we increase F to F + 1, 

 and Z to Z + 1; and so with the other coordinates. In 

 the first case, the sum of the remainders a, /3, 7, is 1, and 

 there is one seat to be allotted to a remainder; in the 

 second there are two seats to be allotted. 



30. Returning to § 26, in which we saw that the 

 seats may be considered to be allotted as if the strength of 

 party A were p' in place of p; the strength of B, q' in 

 place of q; and the strength of C, r' in place of r, let us 

 first examine what will be the solution if we consider the 

 apportionment to be ideal when the differences between 

 p and p , q and q , r and r' respectively, are as small as 

 possible; that is, when — 



S(;/ - p)''= P . . . ,• . (5) 



is a minimum. 



31. Putting p = xQ, &c., and substituting from (2), 

 we get — 



For the minimum value of (5), ^^ = 0, and x = 



X 



o> 



2/ =" 2/oj -^ = ^0' ^^t i^ ^^is case it will not usually be pos- 

 sible to satisfy the further condition of the problem that 

 X, y, z shall each be an integer. 



For other values of k^ (6) is a sphere having its centre 

 at {xq, 2/0, 2o)> or ^^6 ideal point 7, and intersecting ABC 

 in a circle whose centre is I. As k"^ increases from 0, the 

 sphere expands from the ideal point, and is cut by ABC 

 in a gradually increasing circle. 



32. We now have the solution of the problem. In the 

 triangle ABC, plot the point / whose trilinear coordinates 

 are {x^, y^, z^). If x^, y^, z^ are integers (that is, if the 

 strength of each party is divisible without remainder by 

 the electoral unit), / will be a node of the lattice and the 

 solution is {x^, y^, z^). 



If Xq, y^, Zq are not integers, we have to choose x, y, z 

 so that k^ is a minimum. The minimum value of k^ is 

 that for which the gradually increasing circle of inter- 

 section of ABC with the expanding sphere (6) passes 



