146 KEPOET— 1889. 



given, the formula is still the same, viz., Volume in year x oc Value in 

 year x -;- Index-nnmber indicating ratio of the level of prices in the 

 year x to the standard level. The only difference is that we must now 

 base our Index-number upon a part only of the total trade whose volnme 

 is required, assuming that what is true of a part is true of the whole. 



How now are we to determine the <(/pes which enter into our formnla ? 

 First as to the quantities. The most obvious course is to take the g's 

 of a particular year for the typical y's, e.g., q^.^ for •/„, and so on. In the 

 absence of special reasons in favour of or against certain years, we may 

 select any one of the n years to furnish the typical quantities. We have 

 thus at once ?«• different schemes. But it need not be postulated that the 

 same system of quantities should be adopted for each of the series of 

 years. In fact, in the scheme of Mr. Giffen's Table IV. different factors 

 are employed for each comparison, namely, the factors furnished by each 

 year which is being compared in respect of its level of prices with the 

 standard year (1861). If this additional liberty is used to its full extent 

 for every one of the n schemes already enumerated, we have now n 

 variants ; that is, in all we have n^ different formute. However, it may 

 be admitted that these additional schemes, with the important exception 

 of the particular one used by Mr. Giffen in his Table IV., are, if not less 

 accurate, at least less elegant than those which were mentioned first. 

 We shall therefore dismiss these variants with the exception of that 

 one which seems peculiarly appropriate. So far, then, we have (?i + l) 

 schemes presented by the varieties of the quantity-types, the price-type 

 being supposed fixed. 



But the price-types also are manifold. A system of such types is fur- 

 nished by the actual prices of every year — in the absence of special reasons 

 against some particular year. Thus Mr. Giffen has chosen 1861 as the 

 year of standard prices, Mr. Bourne 1883. We have thus n additional 

 cases, which, compounded with the (w-f 1) above ground give ?i (71 -J- 1) 

 distinct schemes or formulEe for comparing the series of volumes. 



Out of this whole number there are 2n which deserve particular atten- 

 tion, namely, those in which the quantities or factors employed in each 

 comparison are supplied by one of the compared years. One system of 

 such schemes in number n is obtained by using in every comparison 

 the factors supplied by the year of standard price ; in other words, by 

 taking the types of price and quantity from the same years. The other 

 system, also numbering n, is that which was noticed in the last paragraph 

 but one as having been used by Mr. Giffen in his Table IV. The quan- 

 tities in this system are supplied by the year which is being compared in 

 respect of its level of prices with the year of standard price. Of course, 

 if we were concerned with only one comparison at a time, if each comparison 

 were an independent operation, these selected schemes would be entitled 

 to a decided preference. But where the object is to find a series of 

 numbers, representing by the ratio of any one to any other the proportion 

 between the volumes of trade for the corresponding years, there seems to 

 be no advantage in constructing our measuring-rod with the factors of 

 one year rather than another. The whole computation presupposes some 

 such hypothesis as that which has been enunciated above ; and on that 

 hypothesis one year has no claim to be preferred before another. 



What may be said in favour of the selected schemes is that they are 

 very slightly more convenient than the other ones. In general, it may be 

 observed that we have n operations, each of a kind illustrated by the for- 



