180 
Miscellanea 
Then iD^ = iDx-bT-cT{ti + 1^) -dT {t{^ + tit., + ti^) - etc. 
, Z)^. = 1 - {i T+ cT^ + dT^-^ etc.} - [ir T+ MT^ + ^eT'^ + etc.} 
- {Sc^r + ficTHetc.} - etc. 
In this series the coefficients of t^, to, etc. are all constants and the highest power of (i is one 
lower than before, so that by repeating the process again and again we can eliminate t from the 
variable on the right-hand side, provided of course that the series ends at some power of t. 
When this has been done, we get 
= )i-0 A' constant, 
„Z)j, = iiDy 4 a constant, 
and of course r ^^2) ^jD^~^ D T)y^ ^'^^ "^j; ^^^^ ""^^ random variables independent 
of time. 
Hence if we wish to eliminate variability due to position in time or space and to determine 
whether there is any correlation between tlio residual variations, all that has to be done is to 
correlate the 1st, 2nd, 3rd...wth differences between successive values of our variable with the 
1st, 2nd, 3rd.../(th differences between successive values of the other variable. When the cor- 
relation between the two ?ith differences is equal to that between the two (n + l)th diSerences, 
this value gives the correlation required. 
This i^rocess is tedious in the extreme, but that it may sometimes be necessary is illustrated 
by the following examples : the figures from which the first two are taken were very kindly 
supplied to me by Mr E. G. Pcake, who had been using them in preparing his paper "The 
Application of the Statistical Method to the Bankers' Problem " in The Bankers' Magazine (July — 
Augu.st, 1912). The material for the next is taken from a paper in The Journal of Agricultural 
Science by Hall and I\Iercer, on the ei'ror of field trials, and are the yields of wheat and straw on 
500 acre plots into whicli an acre of wheat was divided at harvest. The remainder are from 
the three Registrar-Generals' returns. 
I 
II 
III 
IV 
V 
VI 
Correlation between ... 
and 
Sauerbeck's 
Index numbers. 
Bankers' Clear- 
ing House 
returns per 
head 
Marriage 
Kate 
Wages 
Yield of 
Grain 
Yield of 
Straw 
Tuberculosis Death Bate. 
Infantile Mortality 
■ 
Ireland 
England 
Scotland 
Raw figures 
First difference 
Second difference ... 
Third difference ... 
Fourth difference ... 
Fifth difference ... 
Sixth difference 
- -33 
-I--51 
-1- -30 
+ •07 
+ •11 
+ -05 
- -52 
+ •67 
+ •58 
+ •52 
+ •55 
+ •58 
+ •55 
+ •753 
+ •590 
+ •539 
+ •530 
+ ^524 
+ •63 
+ •75 
+ •74 
+ •35 
+ •69 
+ •74 
+ •02 
+ •51 
+ •65 
Number of cases 
41 years 
57 years 
500 
plots 
42 years 
The difference between I and II is very marked, and would seem to indicate that the causal 
connection between index numbers and Bankers' clearing house rates is not altogether of the 
same kind as tliat between marriage rate and wages, though all four variables are commonly 
taken as indications of the short period trade wave. 1 had hoped to investigate this subject 
more thoj-oughl^' before publishing this note, but lack of time has made this impossible. 
