Beatrice M. C^avk and Karl Pearson 
347 
think in the relative largeness of m compared with 71 (i.e. at a maximum 6 as 
compared with 28), rather than in our not having taken sufficiently high 
differences. 
We now turn to the correlations. These are given in Table IV, the actual 
values of the standard deviations of the quantities and their differences being 
recorded along the diagonal cells, while the other cells contain the correlations of 
each pair of variates and of their successive differences. 
We will now consider these correlations in detail. 
(a) Synthetic Index (Arithmetic Mean) with Individual Indices. 
We see at once that the Synthetic Index is highly associated with Shipping 
(> "85), with Importation of Goal (? > '75), and International Commerce (? < '68), 
and fairly highly with Revenue (c. 'bb). On the other hand the sixth difference 
correlations with Post (c. '15), with Stamp Duties (c. '24) and with Savings (> "23) 
are all such (i) that they might well have arisen from the spurious element in the 
Synthetic Index correlations, and are all less than their Andersonian (steady 
value) probable errors. Almost the same may be said of the Railway Index ; it 
is not beyond suspicion of being spurious, and is scarcely significaut having regard 
to its probable error. The Consumption of Coffee is also not very closely associated 
with the Synthetic Index ; it is only about twice its probable error (■4<27 ± "205), 
and a good deal of its value may be spurious. Further in the case of both 
Coffee and Railways, the correlations are still falling between '04 and "Oo for each 
difference. The last individual index remaining is that for Consumption of 
Tobacco and although the correlation of sixth differences is not really significant 
it is negative (— -247 + "2.35), and is exhibiting a steady negative rise. 
Stripped therefore of the common time factor the Synthetic Index will be seen 
to be no very appropriate measure of trade, business activity, and spare money for 
savings and luxuries. With Post, Stamp Duties and Savings, it has probably only 
a spurious relationship, expenditure on railways has little influence, that on 
luxuries is very slightly significant, or indeed in the case of tobacco negative. It 
is, however, closely related to variations in external trade, i.e. imports (including 
coal), to exports and shipping and to effective revenue. It appears to us that a 
suitable general index of prosperity, which will distinguish between a continuous 
growth in all factors with the time, and favourable and unfavourable fluctuations 
from this growth, can only be obtained, when there has been far more ample 
study of the associations of individual indices among themselves, and of these 
indices after they have been freed from the time factor, i.e. of associations 
between high difference correlations. From this standpoint the study of General 
Index theories is at present in its infancy. 
(b) Railway Index. This index is very noteworthy in the nature of its 
associations after removal of the time factor. We have reached a steady 
correlation (c. "62) with Shippi7ig, but beyond this no values of first class im- 
portance appear. The relation of Railways and Revenue after falling practically 
