in the theory of value cuid prices. 69 



of the first derived surface ; i. e., will be the marginal utility of A^, 

 while the right and left slope will be the marginal utility of B^ or 

 the ordinate of the second derivative surface. The primitive surface 

 thus supplies a convenient way of uniting in thought the two mar- 

 ginal utilities. Its absolute height* above the plane of the paper is 

 of no consequence ; it may be lowered or heightened without dis- 

 turbing tangential directions or affecting its two derivatives. 



The three surfaces thus constructed need not extend indefinitely 

 over the plane. They may approach vertical plane or cylindrical 

 asymptotes so that for some points in the plane there may be no 

 surface vertically over or under. 



Mathematically the total utility and marginal utilities at these 

 jDoints are imaginary. Economically it is impossible that the indi- 

 vidual should consume quantities of (<:i) and {h) indicated by the co- 

 ordinates of such points. f Those parts of the plane where such 

 points are may be called " empty." 



If (fig. 18) the point P moves vertically (up and down on the page) 

 the extremity of the perpendicular for the total utility describes 

 one of Auspitz und Lieben's curves for A^, it being understood how- 

 ever that the quantities of other commodities do not change. J; 



The perpendicular for the marginal utility of A^^ generates in the 

 first derivative surface a JevonianS curve of utility for A, it beino; 

 understood that B^, C^, etc. are constant. This curve loill usually 

 descend but it may not and cannot in certain regions if the surface 

 is derived from a primitive with two maxima, or any concave primi- 

 tive. The other perpendicular, however, traces a curve which has 

 never been used, viz : one which shows the relation between the 

 quantities A^ and the marginal utility of B^ ichile B^ remains con- 

 stant. This curve will in general descend or ascend according as 

 the articles {a) and [IS) are competing or completing. For instance, 



* It is in fact the arbitrary constant of integration. 



fThis "asymptote" and "imaginary" interpretation appears to cover the 

 class of difficulties which led Marshall to say his curves failed to have meaning 

 at points at which the individual could Mot live. 



X\\, is rather, then, an " Elementarkurve " of a " Lebensgenusskurve " there 

 being an " anfangsordinate." 



§ Jevons' curve is evidently the derivative of Auspitz und Lieben's. See table 

 Appendix I, Division II, i^ 3. 



