88

Irving Usher—Mathematical investigations

and distribution. Definition 3, Part I, Chap. I, § 4 yielded uniformresults only on the assumption that the utility of each commodity wasindependent of the quantity of others. Similar assumptions are nec-essary in geometry. A unit of length is a yard. A yard is the lengthof a standard bar in London . To be used it must be assumed that itslength is not a function of its position nor dependent on the changesin length of other bodies. If the earth shrinks we can measure theshrinkage by the yard stick provided it has not also shrunk as a nec-essary feature of the earth’s change. Definition 3 was essential inPart I to give meaning to the cisterns used. Such a definition is essen-tial to the analyses of Gossen, Jevons, Launhardt, Marshall, and allwriters who employ coordinates. Yet it is not necessary in theanalysis of Part II.

§ 5 .

In fig. 28 the “ lines of force ” are drawn perpendicular to the in-38 . difference loci. The directions of these lines of

force are alone used in the formulae in Ch. II, § 9which determine equilibrium. Therefore thedirections alone are important. It makes abso-lutely no difference so far as the objective de-termination of prices and distribution is con-cerned what the length of the arrow is at onepoint compared with another. The ratios of thecomponents at any point are important but theseratios are the same whatever the length of thearrow. Thus we may dispense with the totalutilitv density and conceive the “economic world” to be filledmerely with lines of force or “ maximum directions.”

§ 6 -

Even if we should give exact meanings to the length of these ar-rows (so that the equation vU, = F(I) should signify not only thatfor each position in the economic world a definite “maximum direc-tion ” exists but also that the rate of increase of utility or the lengthof the vector along this line is given)—even theii there would not bea complete primitive U 1 = 9 ?(I) unless certain conditions were ful-filled.* These conditions are (1) that the lines of force ai’e so ar-ranged that loci (surfaces in two dimensions, rn — 1 spaces in m di-mensions) perpendicular to them can be constructed, and (2) that

* Osborne, Differential Equations, p. 12.