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



1913. 



whence it can be shown that — 



ni- 



p V 



The quantity to be made a minimum is, then, S (^^'V/^). M, 

 Sainte-Lague suggests a geometrical solution similar to 

 the method used by me for three parties, but gives the- 

 following practical solution for any number of parties. 

 The identity — 



^■- = 1 + 3 - 5 + ... -r (2a - 3) -f (2.r - 1) 



shows that the sum to be made a minimum is the sum of 

 the X first numbers of (1), the y first numbers of (2), &c. ; 

 X, y, z being chosen so that the m smallest numbers are- 

 selected : — 



(i> 



(2). 



As the same result would be obtained by inverting all 

 these numbers and choosing the m largest we have the 

 following rule: — 



Rule of least squares: Divide p, q, r ... by the odd 

 integers 1, 3, 5 ..., and in the various series of quotients, 

 so obtained select the largest, until m have been obtained.. 

 Party .4 receives as many members as the number of 

 quotients taken from its series; and so with the other- 

 parties. 



57. Next, consider only positive errors {i.e., errors for 

 electors who are over-represented). If the error for each 

 elector of party A is positive, this party has at least 

 X + 1 seats {X + 1 being the whole number next greater 

 than X^^). According as the seats obtained are X + 1., 

 X 4- 2, ..., the error for each elector of .4 is^ — 



JC + 1 m X + 2 m 



p 



V V V 



and so for the other parties; and the remaiaing seats have- 

 to be allotted so that these errors may be as small as pos^- 

 sible. If the parties have obtained X, 7, Z . seats, and 



