XLIV] Introduction <-r\i 



Two points may be noted in connection with the Table: 



2 

 (a) The heavy type function is always greater than 4 =-; it is then 



idle to seek for values of e, e', e" unless the ratios on the left of equations (iv), (v) 



2 



and (vi) arc greater than 4 -- -= . If one or more of them be less than this value, 



n + 1 



either we have not taken high enough differences, or the X 9 , F,'s in regard to their 

 intercorrelations do not even approximate to a simple law of geometrical decadence. 

 (6) It may also be that the ratio of the second moments or the product- 

 moments of successive differences lies outside the possible values of the e-functiona, 

 and again a geometrical decadence of the X v and Y 9 correlation cannot hold even 

 approximately. 



Illustration (i). In dealing with Italian Economic Indices*, only those for 

 Gross Receipts of Railways (Rail), Importation of Coal (Coal) and Return of 



Imported Coffee (Coffee), provide values of a^&pxl^tfx which exceed f 4 - TTli 



or since n = 5, exceed 3'667. The values of these ratios are: Rail, 3711; Coal, 

 3'G82 and Coffee, 379Lf. Looking out these values in the column marked n = 5 

 in the Table we obtain by rough linear interpolation 



Rail, / = - '2135 ; Coal, e a ' = - "0809 ; Coffee, e 8 ' = - '4941. 

 We see accordingly that there is a difference between the second and the first 

 and third indices. In Rail and Coffee a rise in the index deviation above the 

 secular value in one year is correlated negatively with the index deviation in the 

 following year. In the case of Coal a rise in the index deviation above the secular 

 value in one year is also correlated negatively but hardly significantly with that of 

 the following year. In other words if more Coal in excess of the secular value be 

 imported in one year, that excess will hardly be associated with a defect in the 

 following year. An immediate balancing, or see-saw effect, is sensible in Railway 

 Receipts and in Imports of Coffee, where the swing above the secular value is 

 sufficient to give a significant negative value on the average to the correlation 

 between successive years. 



We now determine from these values of e' the corresponding values of the small 

 type function ; they are for 



Rail, 1-404,322 ; Coal, 1-142,834 ; Coffee, 2'104,922. 



But these will be of no service unless the ratios of difference products are con- 

 sistent with geometrical decadence. 



We should accordingly find the ratios 



* Sec G. Mortara, Giornale degli Economitti, Feb. 1914, and for the computations of tlie constants 

 cited, Itiometrika, Vol. x. pp. 34G 349. 

 t See Biometrika, Vol. x. p. 34(1. 



