16 QUOTA IN PROPORTIONAL REPRESENTATION, II., 



R.S. TAS. 



List Systems. 



54. In §§ 22-25, I discussed, with the aid of the method 

 of least squares, the problem of partitioning a given num- 

 ber of seats among three parties in proportion to their 

 strengths. I had not then seen any mathematical discus- 

 sions of this problem ; but I have since obtained a copy 

 of a paper published in 1910 by M. A. Sainte-Lague, Pro- 

 fesseur de Mathematiques speciales au Lycee de Douai 

 (and now of Besan9on), in which he gives a discussion of 

 the problem for any number of parties by the same 

 method. (-^) The volume in which this paper is published 

 is not accessible to many students in this part of the world, 

 and I have therefore made a summary of M. Sainte- 

 Lague's results. 



I use the notation of § 25 of my own paper, as corrected 

 in the erratum slip. 



55. Each elector, says M. Sainte-Lague, has the right 

 to be represented by a fraction of a deputy given by 

 ml V = 1/ Q. If he belongs to the party A, he is repre- 

 sented by the fraction x/]} of a deputy; whence the error 

 in representation for him is seen to be x / jj — m/v. For 

 the electors of the various parties, there are errors 

 «i> ^2» ^3' •••^s. The various methods diverge from one 

 another in the ways in which they endeavour to make these 

 errors as small as possible. 



56. To arrive at the best rule, M. Sainte-Lague applies 

 the method of least squares. 



For each elector of party A the error in representation 



X m 



The sum of the squares of the errors for the p electors 

 of this party is — 



'% - v) 

 and the sum of the squares of the errors for all v electors 



(*") La irp resfi)itation J) rojwrtionnelle et la methode des moindres carres. 

 Annates Scientifiques de rEcole Normale Sup«irieure, He. serie, tome 27, 

 December, 1910, pp. 5:30-542. M. Sainte-Lague has given a more popular 

 account of his results in the Revue Generale des Sciences pares et appliquees 

 of 30th October, 1910, pp. 846-852 ; and the " rule of least squares " is 

 stated in a communication made to the Academy of Sciences of Paris on 

 8th August, 1910. 



