BY E. L. PIESSE, B.SC, LL.B. 51 



party lies between 2/3 x (100 - (m - 2).100/m) and 

 3/2 X 100 /m, it may obtain two members, although 

 entitled only to one. The range of strength for which dis- 

 proportionate representation is possible is therefore — 



2m — 1 100 2 f 100\ 



m m 



- (lOO - („, - 2) — ) 



Proceeding in the same way for the cases in which the 

 smaller party is entitled to three or more members, we 

 obtain the following expression lor a measure of the range 

 of strengths for which the Hare quota may give dispro- 

 portionate representation : — 



r 100 I ^ 100-1 



L/w -f 1 2 m ^ 

 r2m - 1 100 



2 / 100\-1 



L m m 



■Sm - 4 100 3 / . 100> 



r6m — 4 100 3 / , 10Un-| 



L w — 1 m 4 V ^ m /J 



1 



m + \ — m 



Ul \ 100 m - 1 / 

 (m - 1) - ) - • ( 100 - 



100x-| 



[ 



m 100 2m — 1 lOOn „. 



* — * I • • • \ " ) 



m + [ m 2 m J 



All such strengths are shown on the graph by the lines 

 marked '' H " connecting arrow-heads. 



17, With the Droop quota the following expression gives 

 a measure of the range of strengths for which there must 

 be disproportionate representation : — 



100 1 100\ /, 100 3 100 



/ 100 1 ]m\ / 



Iw + 1 ~ 2 * m ) ^ \^ 



6 iK}V\ 



77Z + 1 *^ m J 



/ 100 5 lOOx . 



+ ( 3 • r - i^- • ) + ...+ 



V w + 1 2 m J 



(i \^ 1^0 2m - 1 100\ ... 



V^ ' m -\- \ 2 w / ^ 



All such strengths are shown on the graph by the lines 

 marked '' D " connecting arrow-heads. 



