in the theory of value and ^jrices. 91 



--- ) 



separate equations but only n[^m — V) are independent. Which shall 

 be selected is a matter of convenience. We may make ever3^ equa- 

 tion contain j9„^^ for instance and write 



' __ d\5 d\S _ d\5 d\5_ ___ 

 P.,^ : p,,^ - : ^^g _ -^-^^ : ^-- - etc. 



77)1 77)1 7r>2 77)2 



Pa,n : Pc,rr = ^tC. 



Pa,n • Pa,K -— > 



Pa,Tr • l^a,e ~- > 



Now from the first two equations we may derive by division 



*_ _c?U_ d\J 



but we might wish to use this last as one of the n('37n^l) indepen- 



dent equations, if Aj^^ should "fail." From the 7i{3m — 1) — sepa- 



rate equations we are at liberty to select for use any 72(3w — 1) inde- 

 pendent ones; and if in this selection there occur any which by some 

 change of quantities fail, we are compelled to change our selection so 

 that the new n{37n — 1) equations shall avoid the "failing" magni- 

 tudes. 



This is interpreted in the mechanism as follows : when a cistern 

 is wholly above the surface of the tank (as IIIC fig. 8) and so con- 

 tains nothing, the quantity of commodity and its utility " fail." The 

 levers which keep the ordinates in proportion to the corresponding 

 ordinates of other individuals may be far more numerous than the 

 levers pictured in former diagrams. Thus for four cisterns there 

 may be six levers (by joining each pair) but only three are neces- 

 sary. The " failure " of any magnitudes will not invalidate any 

 system of levers originally selected ; it will simply make their num- 

 ber greater than necessary. 



II. THE CISTERNS AND DIAGRAMS OF PART I COMPARED 



WITH THE DIAGRAMS OF JEVONS AND OF 



AUSPITZ UND LIEBEN. 



§1. 

 In order to represent geometrically the relations between quantity 

 of commodity, marginal utility, total utility, and gain (any two of 

 which four magnitudes are determined by a specified rehxtion between 



