Page:Carroll (1884).djvu/29

 Also let it be assumed that an Elector may not give 2 votes to the same Candidate. (N.B. 'cumulative' voting is discussed at p. 27.) Now, in order that x may be sufficient to fill s seats, it must be large enough to make it impossible for the other (e − x) Electors to fill (m + 1 − s) seats; since the two events are incompatible, so that, if the latter were possible, the former would be impossible. To effect this, each of the s Candidates must have more votes than it is possible to give to each of (m + 1 − s) rival Candidates.

In order that x may be necessary, it must be only just large enough for the purpose.

It will be necessary to consider the following 4 cases separately. Observe that > means 'greater than,' ≯ means 'not greater than,' and ∴ means 'therefore'.

Case (a)v is ≯ s, and also ≯ (m + 1 − s); Case (b) > s but ≯ (m + 1 − s); Case (c) ≯ s, but > (m + 1 − s); Case (d) > s and also > (m + 1 − s).

In case (a), the x Electors can give vx votes, which, divided among s Candidates, supply them with $vx⁄s$ votes apiece. Similarly,