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



65. 



/?, 5^, r , the strengths of the several parties 



A, B,G , . . . 

 V, the total of all the votes polled. 



Jo, 2/0 J 2^0, ... , numbers proportional to ^, q,r 

 and such that — 



^0 + 2/0 + 2o + ••• = ^ • • . 



Q^ the electoral unit, given by — 



m 



V 



m 



(1) 



{n 



. . , the integral parts of x^, y^, Zq 

 the fractional parts of .r^, i/q^ Zq . 



X, Y, Z 



a» 01 7» • 

 so that — 



i'o - A" = a, //o - y = fi, z 



-Z = 



(3) 



p', q' , r . . respectively equal to XqQ, yf,Q, ZqQ . . . 

 2', //, 2 . . . the seats obtained by the parties with the 

 method of apportionment actually used. 



26. The apportionment would be ideal if the rule-of- 

 three method could be used without allotment of seats to 

 remainders ; that is, if the seats obtained by the parties 

 were Xqj y^, Zq . . . To allot Xq, 2/o) ^0 . • seats to the 

 parties is equivalent to taking their strengths to be 

 V i 9''> ^' • instead of the actual p, q, r . . . 



27. Confining ourselves to three parties, we have for 

 Xy y, z the equation — 



X + y + z = m (4) 



This is the equation of a plane which cuts the axes of 

 a;, y, z, at points A, B, C equidistant from the origin. 

 As X, y, z are positive, the only portion of the plane to be 

 considered is that in the octant in which all the coordinates 

 are positive; this portion is the equilateral triangle ABC, 



Fig. 2. 



