148 EEPORT — 1889. 



mean year with normal or typical prices formed by arranging tlie actual 

 prices of several years. This principle has been employed to some extent 

 by Jevons, Dr. Soetbeer, Mr. Palgrave, and Mr. Sauerbeck. In virtue of 

 this principle the n different bases of price which we found before are 

 swelled to 2"— 1. Altogether, therefore, we have 2"x(2" — 1) distinct 

 schemes of Index-number. 



This account may be further multiplied if we have a choice, as would 

 often be proper, between the Arithmetic Mean and a certain other species 

 of average which is noticed below. However, it may be well to leave 

 some margin for the occurrence of abnormal years (like 1873) whose data 

 cannot be used freely. So let us be content with the modest estimate 

 just furnished, as resulting from that degree of liberty of choice which 

 we have so far contemplated. 



That is, however, a very narrow view. For each q which we have 

 employed may be replaced by an expression which is by hypothesis of the 

 same order. For instance, we are entitled to put for q^,j, q,^y, &c., the 



expressions g„y^', qi,,—, &c. ; say q'a,j, q'h,j, &c. This is, in effect, what 



Pax Vbx 



Mr. Giffen has done in the classical computations comprised in the first 

 thi'ee tables of his reports. His formula for the volume of any year, z, may 

 be written in our notation : — 



(?«75|^)Fa.+ {qm:^^Pi.z+ Ac- 

 Volume of year zoc Value of year z-i- ~ —^ ^-^^ 



{?«r5^Mi5«Cl + ((^675^MP661+ &c. 



The reader will easily see the equivalence of this formula to that which 

 Mr. Giffen has made familiar, if the symbols p^^^, p^gj, &c., are brought 

 outside the brackets both in the numerator and denominator. The de- 

 nominator, for instance, will become (qa7oPa75) + (9'675 25w3) + &c. ; corre- 

 sponding to the column headed 1875 in Mr. Giffen's Table II. 



By parity it may be shown that the Index-number constructed by 

 Mr. Palgrave implies the following formula for volume : — 



Volume of year z oc Value of year " '' 



(?-f^) X '^a+ (^'^ ~) "^ + &°- 



where tt^, ttj, &c., are types of price obtained by taking an average over 

 certain years.' In fact, Mr. Palgrave's scheme may be regarded as a 

 variant of the plan employed in Mr. Giffen's Table IV., which was above 

 commended to particular attention. 



These ^''s may be combined with each other in the same way as the 

 q's ; and, indeed, the q"s and the q's may be mixed. However, these 

 operations would be laborious and inelegant. We shall, therefore, cull 

 from the infinite field which has just been opened up only just such a 

 number as to double the estimate already reached. It may be useful to 

 show how this additional contingent is reached, taking as a conspicuous 

 instance the materials of Mr. Giffen's work. It was open to him to have 

 taken for the basis of prices some year other than 1861. In fact, in his 



' See, on Mr. Giffen's and Mr. Palgrave's Index-numbers, sect. ii. of the present 

 writer's first Memorandum, Brit. Assoc. Jlej)., 1887, p. 265. 



