14 Flux, Fall in Prices during the past Twenty Years. 



If also we write P n = 7r n a n , ir n denoting the price-level, 

 a n the quantity, we have 



Tn n _ Pn X n 



7T n _i i? n Pn-1 



and 



a n _i P n -i - a n _ l x n -i 

 In practice — — is not ascertained, and — — is used in 



its place ; also ^-is not ascertained, and ^ n * n is used 



a n _i # n -l # n -l 



in its place. In contrasting the n th and the r th years we 

 have 



• 



TTr A JV^n-l J/r+1 



a n== a n x n y n -y n -i 



a r <2 r # r ^n-^n-l ^r + 1 



In using the proportions of the p's and a's in place of 

 those of the 7r's and a's therefore, we are liable to intro- 

 duce errors. 



It is clear that 



7T r a r # r # r 



which might have been arrived at without using the 



values above shown for — and — Thus when using, 



in place of the true ratios, the values of the calculated 

 price- and volume- ratios, errors might exist in both 

 the latter ratios as estimated, but not in their product. 

 Only if each x be equal to the corresponding y, could 

 we get accurate results for each of the ratios separately. 

 This last condition implies that the price-variation is 

 the same for the quantified and unquantified parts of the 

 total dealt with. 



