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



If also we write P n —ir u a m ir n denoting the price-level, 

 a n the quantity, we have 



and 



O^n-l P n-1 - a \\-\ x n-l 



In practice — — is not ascertained, and - 1 - is used in 



7Tn-l Pn-l 



its place ; also — —is not ascertained, and — - — - is used 



a n _i ^ n _i x n _i 



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

 have 



T^n Pn ^iT^n-1 ^r+ 1 



7T r ~pv ' yn-yn-1 J'r+l 



a n = a n x n ^ y n .y n -i y v +i 



a r a r x v x n .x n _i ^r+i 



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

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

 duce errors. 



It is clear that 



7Tn« n_ A • <Vn 

 7T r a r ^ r fl r # r 



which might have been arrived at without using the 



values above shown for — - and — - Thus when using, 



7r r a r ' 



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. 



