94 Irving Fisher — Mathematical investigations 



For the new coordinates the substitutions from the table give : 



fxdy-\-qy = m, 

 which reduces to cc = — q^ 



a straight line parallel to the axis of ordinates (fig. 37). 



The Auspitz und Lieben curve does not reveal to the eye the spe- 

 cial supposition (that commodity and marginal utility change pro- 

 portionally). If we suppose that marginal utilitj^ decreases at a con- 

 stant rate in relation to constant second differences of commodity, 

 the new diagram reduces to a straight line : 



x — qy-^m z=z 0, 



while the other curves would be : 



(y«-fAa;.+B)^ = C(D~a3j« 

 and a?^- =: E — F^/^— Gy/. 



§4. 



The value of Jevons' diagram consists in the use of a simple and 

 familiar system of coordinates (the Cartesian) as representing the 

 two chief economic quantities, and is probably the best for elemen- 

 tary purposes. 



The value of Auspitz und Lieben's diagram together with a 

 ^' derivative " curve* not shown above consists chiefly in the ease with 

 which maxima are discovered and the clear association of maxima 

 with equality of marginal utilities. It is believed that the third 

 method will, by means of its applicability to the mechanisms of 

 Part I, more clearty reveal the interdependence of the many com- 

 modities of many individuals and of their many utilities. 



§5- 



The properties which are essential to the curve we have adopted 

 are: — 



First. That the curve shall never admit of being intersected twice 

 by a horizontal line (i. e. that it shall not cease to run in a general 

 up and down manner), to express the fact for consumption that mar- 

 ginal utility decreases as quantity of commodity increases and for 

 production that marginal disutility increases as the quantity of com- 

 modity increases. 



* Whose Cartesian coordinates are Xa and i/a — — 



dXa 



