Ethel M. Elderton and Karl Pearson 
495 
If our first variable be taken* as a; = 0i (t) + X, where X is the intrinsic vahie 
of a; as apart from the time change, then mean after steadiness has set in is 
* One of the bases of the variate difference correlation method lies in the assumption that the 
intrinsic variation is superposed on a secular change of a continuous character ; the causes which 
determined the intrinsic variation X are supposed to be sensibly independent of the time for the 
period under consideration. We conceive the secular change as given by a parabola, say, of the 
sth order, but the deviations from this curve are supposed in magnitude and sense to be independent 
of the time, i.e. due to chance causes which are the same in 1850 as in 1900. This assumption 
is an important one and must lead to our seeking relatively short periods consistent with a numerical 
frequency sufficient for significance. It can be roughly tested, of course, by considering <jx as found 
from, say, the first and second halves of our observations. In our own case we found : 
Values of deduced from Sixth Differences for \st 25, for 2nd 25, 
and for all 50 years. 
1* 
{'«3) 
$ 
? 
? 
$ 
? 
$ 
? 
$ 
? 
1st 25 years ... 
All 50 years ... 
2nd 25 years . . . 
7- 32 
8- 61 
9- 70 
6- 94 
7- 83 
8- 61 
5-51 
4-71 
3-73 
5-Gl 
4-63 
3'37 
2-09 
1-59 
0-83 
2-30 
1-77 
0-98 
1-52 
117 
0-66 
1-67 
1-28 
0-68 
1-05 
0-86 
0-63 
0-91 
0-78 
0-G2 
These values are less steady than we had originally hoped for. Clearly the variability of the X 
portion of the infantile deathrate has grown greater, and that of the four child deathrates has grown 
sensibly smaller with the time. The fundamental hypothesis of the variate difference metliod is there- 
fore only approximately true for this material. We have made some investigations on the assumption 
that x = <px[t)-\-{a-\-ht)X, but the values of a and h obtained were by no means satisfactory. We have 
in hand a further investigation of the problem by the method, originally suggested by one of us, before 
the difference method was started ; namely to subtract from x the value obtained by the best fitting 
parabola of the sth order in the time and so to reach the actual values of A'. The relation of these 
to the time can then be found with some degree of accuracy. To the male deathrates of the second and 
fourth years of life we applied parabolae of the third order in the time, and obtained excellent fits ; we 
then subtracted the ordinates of these parabolae from the deathrates and correlated the remainders, 
^2 and (^4 say. We found '■^_^^^= +"312 ±-088, a value corresponding more nearly with g.,,,^ than 
''Sa??t) 53m4' indicating that we might more rapidly approach final values by this method than by 
that of variate differences. But the fitting of high order parabolae is very laborious; at the same time 
the graphs give excellent tests of the accuracy of the work, and we obtain the actual values of what 
we have termed A' and Y, as represented by d-i and d^. We then correlated the numerical value of rf^ 
with the time and found r^^ ^= --284 ±-089. It is clear that with correlations of this order with the 
time, r ^ ^ would not be modified by the extent of its probable error if we found the partial corre- 
lation f^i^^^i or corrected the correlation of d-, and J4 for the time. There is another point, however, 
which justifies us in disregarding this variation of A' and Y with the time as of secondary importance. 
The correlation of X with the time i% positice in the first year's mortality and negative in the following 
four years ; thus while it would certainly tend to give a negative value to for the 1st and 2nd 
years of life, it would tend to give a positive value to the correlation for all successive pairs of years 
beyond the 1st and 2nd. Now all such successive pairs of years have high negative values, which are 
therefore minimum values, but these values are all in excellent agreement — roughly equal to - -7 — with 
that found for the 1st and 2nd years of life. We therefore concluded that the influence of the time on 
the deviations from the secular curve of change, although very sensible, is of no substantial importance 
for he correlations. 
