388 
Values of First and Second Prizes 
The values of X between those corresponding to n = 50. and n = 100,000 range 
within a difference of Tl. The smallest possible class in which c is not negative, 
consists of five individuals, and even here the proportion of X to F is as 71 "0 to 
29"0, which does not differ grossly from that in a class of 100,000 where it is 75"4 
to 24'6. Nay, even taking the smallest possible class which is of three individuals, 
in which the values of ha, hh, he are respectively equal to — c, 0, and + c, the value 
of h(a — c) = 2hc and that of h(b — c) = hc. Consequently S = Shc, therefore 
Z - 100 X I, and F= 100 x i, = 667 and 33-3 as in the Table. 
The rationale of the approximate uniformity of the value of X and Y seems 
well worthy of a more searching mathematical investigation than I am competent 
to make. It seems difficult to doubt that this curious property of the terminal 
equi-postiles is associated with others whose character cannot now be foreseen. 
Comparison with facts. Many serious objections present themselves a priori to 
the useful application of this theory, among which is the partial non-conformity 
of examination marks with the law of frequency, especially at either end of the 
series, one of which is precisely the part here in question. I therefore put 
the theory to test by procuring through the kindness of friends a large number 
of sets of marks in various Civil Service examinations. I took them just as they 
came and found the X and Y values for each case, as in the following example. 
No. 268. 
a = 1801 
6=1712 
c = 167l 
a-c = 130 
h-G= 41 
S = 11l 
Z: 100::130:171 ; F: 100 : : 41 : 171 
Z = 78-0; F- 22-0; Total 100. 
I grouped these values into fives, each page of my MS. book containing that 
number, then into twenty-fives, and so on. Individually their values ran very 
irregularly, but the groups of 25 began to give hopeful indications which were 
fully confirmed by larger groupings, as is shown in Table II. where the X 
values alone are entei-ed. Those of F are of course complementary to them. 
Thus far the evidence that the calculation was correct in principle seemed 
conclusive, owing to its being so remarkably well confirmed by observation. In fact, 
I lived for a few days in a fool's paradise, thinking that such was the case, until 
with the desire of probing the matter more thoroughly, I made a Table of the 
distribution of the individual observations. The result is shown in Table III., 
■which shattered my sanguine hopes. If the principle upon which the calculation 
is based had a contributory effect to any noticeable degree, in producing the 
mean value of 73'4 as shown in Table II., there would have been a concentration 
of values about that point in Table III. But there is nothing of the kind. The 
values are pretty equably distributed between 50 and 100, with a slight but 
