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TRANSACTIONS OF THE SECTIONS. 9 
On the Chances of Success or Failure of Candidates for three-cornered or 
four-cornered Constituencies. By KR. B. Harwarn, M.A. 
This paper discussed some of the consequences involved in the system of voting 
now in force for constituencies returning three or four representatives, by which 
no voter can yote for more than two candidates in the one case or for more than 
three in the other. 
Suppose a majority of M voters to bring forward three candiates A, B, C, while 
the minority of m voters bring forward two, and that, of the majority M, x vote 
for B and C, y for C and A, and z for A and B; then, supposing that each of the M 
voters gives both his votes, 
rs pert gt Ad ONE pica ca ome beeen CIs) 
The event of the success of A may be denoted by A, and the corresponding condi- 
tion is y+2> m; the event of his failure by a, and the condition is y+z<m. So 
also a compound event, e. g. the success of A and failure of Band C, may be 
denoted by Adc, and the corresponding conditions are y+z>m, z+r<m, 
zt+y=<m. 
ithe different possible events are then A BC, aBC, 6C A, cA B, Abe, Bea, 
Cab, and the question discussed was the determination of the relative numbers of 
the cases favourable to these different events for different relative values of M and 
m, or the relative numbers of positive integral solutions (including 0) of the above 
equation subject to the different conditions corresponding to the different events. 
It was shown that this discussion was much facilitated by a simple geometrical 
representation. A number of points uniformly distributed over an equilateral tri- 
angle being taken to represent the total number of possible distributions of the M 
votes or the total number of solutions of the equation (1.), each point corre- 
sponding to one solution determinable from its position in the triangle, the numbers 
of points within the different areas into which the triangle is divided by lines pro- 
perly drawn parallel to the sides, represent the cases favourable to the different 
possible events. Thus the relative numbers of these cases are rendered evident 
and easily calculated. 
The figure annexed is drawn to represent the case where m=3M. If each side 
of the triangle be 5, each of the lengths AD, A D', BE, BE’, CF, C F' is 3, and 
A 
B F 5) c 
the seven spaces correspond to the seven possible events marked within them. 
The relative numbers of cases favourable to the different events will be readily 
seen to be 1, 1, 1, 1, 7, 7, 7, if the number of voters be large, so that an area may 
be taken as proportional to the number of points within it. 
Tf each individual distribution of the votes, or, in other words, each solution of 
), were equally probable, these numbers would represent the relative probabi- 
lities of the different events; but this is far from being the case, the weight of each 
