298 
Oh Theories of Association 
or a = 1-0582. Thus a six per cent, increase in a suffices to provide the increased 
blind population at the greater age ; the two curves intersect at x = + 2 - 968er, or 
somewhat beyond the " blind " boundary. For all grades beyond this there will 
be more persons of each grade of bad sight. There is nothing inconceivable or 
improbable in this ; but, on the other hand, there would be beyond x = — 2'968o-, 
on the good sight side, a number of grades of better sight with more people at the 
older age. This, of course, is not impossible, but it seems far more reasonable to 
suppose the average sight to grow worse with old age, and in addition to change 
its variability somewhat. If the variability remained the same, as we have seen 
under the first case, there is no excess in the grades of marked good sight in the 
population of older ages. Let us now consider what happens if the mean be 
shifted and the variability increased. Taking m>M*, we have 
x-m 2- /(m-M) 2 n <r s -2 J . aN er2 m-M 
~a =± ^-SW ~ W~ +2 -^ l08e ^l-a^-~?T ■ 
Now let us take a= 1012, and therefore for the "blind" groups at 45-55 and 
55-65 we have as before X = 2-97402, x = 2*8105<r = 2-83862 to fix the dichotomy. 
Hence 
(m - if )/2 = (X - x)fZ = 13538. 
Thus we find (x - m)/a = + 2*6738 or - 16 2791. 
The former value shows that from some little distance outside the dichotomic 
line each grade of bad sight and blindness has more individuals of that grade at 
ages 55-65 than at ages 45-55. The latter value indicates that the point of 
intersection of the sight curves for the two ages on the side of good sight takes 
place at a point so extremely distant from the average sight that not a single 
individual would occur with such sight in a population many thousand times 
greater than the actual population. 
We think it most probable, however, that a third case, not even referred to by 
Mr Yule, best describes what actually takes place — namely, that the sight at the 
older age gets worse and is less variable, not more variable. To illustrate this, 
take (r = -99X, then X = 2-97402, x= 28105o- = 2-78242. Hence 
(m - M)jt = '19158, and (x - m)/a = + 2"4742 or + 16-5875. 
Thus the older age curve now never cuts the younger age curve of sight on the 
side of good sight at all. It cuts on the side of bad sight twice, once somewhat on 
the good sight side of the " blind " dichotomic line, and on the other occasion 
immensely beyond the limits of the populations in question. In other words, 
the older ages have fewer members in each grade of good sight and more members 
in each grade of very bad sight. This appears to us a perfectly reasonable state 
of affairs, and of course extends far beyond the ratio selected for cr/2. 
It will thus, we think, be clear that had Mr Yule attempted to turn his 
half-baked notions into figures before he expressed tliem in words, he would 
* The positive direction of the variate is towards bad sight. 
