386 
Values of First and Second Prizes 
(1) I concluded that when only two prizes a and /8 are given, their values 
should not be proportioned to the absolute merits of the two competitors, but to 
their respective excesses of merit above the third competitor, who receives no prize 
at all. Let [J.], [iJ], and [C] be the first, second, and third competitors, and a, b, c 
the marks allotted to them, then I conceive the most suitable relation of a to /3 
is as (a — c) to (b — c), and not as a to b. 
(2) If there be n competitors, considered as random samples from a large 
body among whom merit is normally distributed, the most reasonable presumption 
is that they will tend to occupy n equally probable positions. In the ordinary 
table of the Probability Integral the argument is + hx, whose values range from 
0 to + infinity, and the tabular values are those of @{lix) ranging from 0 to + 1. 
For the present purposes @ (hx) must be taken as the argument, running from — 1, 
through 0, to + 1, and hx becomes the tabular value. If there be n competitors 
the most equable, and therefore the most probable distribution of them along the 
scale of + © (hx), is that one competitor should fall into each of the n equidistant 
stalls (J 11 stalls lying on either side of 0), the septa that enclose those stalls being 
situated at 0, 4- 2, + 4, ... + n on the positive side and at 0, — 2, — 4, — on the 
negative side. I assume that each competitor tills his stall, and that his position 
is expressed with needful precision by the middle of the stall. Consequently the 
places of the several competitors will be taken to be at + 1, +3, + 5, . .. + (n — 1) 
on the positive side and at —1,-3, —5, (n — 1) on the negative side. 
Their position is purely a question of evenly distributed probabilities, entirely 
unconnected with the law by which the values of hx to which they refer are 
established. At the same time I am aware that others may hold that this 
method fails in accuracy, by treating the curve of distribution as a polygon, 
but I shall not stop to argue the point further because the difference of result 
is too small to weigh in the present argument. Following a nomenclature already 
adopted, in which the words ' centile ' and 'decile' occur, I will call the n 
values in any array corresponding to those of ®(hx) = + 1, ± 3, + 5, . . . + (n — 1), 
by the name of " equi-postiles," and those of the septa between which they stand 
by tliat of " equi-partiles." 
(3) Thus far it has been implied that the value of ?i is known, but, as a matter 
of fact, it seems usually impossible to arrive at even a grossly approximate idea of 
the number of virtual competitors ; which far exceeds their actual number in all 
important competitions. The number of runners in the Derby are few, but they 
include the best horses out of a multitude of thoroughbreds, who are all qualified 
for entry but whose owners keep them back because their chance of winning was 
found by trial performances to be nil. The same happens in University scholar- 
ships, in the principal athletic sports, and in all competitions that arouse a widely 
felt and keen desire for distinction. 
Therefore being ignorant of n, I selected a few widely different values of it 
for trial and worked out the @ (hx) values of [A], [B], and [G] by the formula 
