in the theory of value and 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, orthe ordinate of the second derivative surface. The primitive surfacethus supplies a convenient way of uniting in thought the two mar-ginal utilities. Its absolute height* * * § above the plane of the paper isof no consequence; it may be lowered or heightened without dis-turbing tangential directions or affecting its two derivatives.

§ 8 .

The three surfaces thus constructed need not extend indefinitelyover the plane. They may approach vertical plane or cylindricalasymptotes so that for some points in the plane there may be nosurface vertically over or under.

Mathematically the total utility and marginal utilities at thesepoints are imaginary. Economically it is impossible that the indi-vidual should consume quantities of (a) and (b) indicated by the co-ordinates of such points.f Those parts of the plane where suchpoints are may be called “ empty.”

§9.

If (fig. 18 ) the point P moves vertically (up and down on the page)the extremity of the perpendicular for the total utility describesone 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 thefirst derivative surface a Jevonian§ curve of utility for A, it beingunderstood that B , C lt etc. are constant. This curve will usuallydescend but it may not and cannot in certain regions if the surfaceis derived from a primitive with two maxima, or any concave primi-tive. The other perpendicular, however, traces a curve which hasnever been used, viz : one which shows the relation between thequantities A, and the marginal utility of B t while B l remains con-stant. This curve will in general descend or ascend according asthe articles (a) and ( b) are competing or completing. For instance,

* It is in fact the arbitrary constant of integration.

f This ‘ ‘ asymptote ’’ and 1 ‘ imaginary ” interpretation appears to cover theclass of difficulties which led Marshall to say his curves failed to have meaningat points at which the individual could not live

Jit is rather, then, an “ Elementarkurve” of a “ Lebensgenusskurve ” therebeing an “ anfangsordinate.”

§ Jevons’ cuTve is evidently the derivative of Auspitz und Lieben's. See tableAppendix I, Division II, § 2.