88 Irving Fisher — Mathematical hivestigations 



and distribution. Definition 3, Part I, Chap. 1, § 4 yielded uniform 

 results only on the assumption that the utilit}'' of each commodity was 

 independent of the quantity of others. Similar assumptions are nec- 

 essary in geometry. A unit of length is a yard. A yard is the length 

 of a standard bar in London. To be used it must be assumed that its 

 length is not a function of its position nor dependent on the changes 

 in length of other bodies. If the earth shrinks we can measure the 

 shrinkage by the yard stick provided it has not also shrunk as a nec- 

 essary^ feature of the earth's change. Definition 8 was essential in 

 Part I to give meaning to the cisterns used. Such a definition is essen- 

 tial to the analyses of Gossen, Jevons, Launhardt, Marshall, and all 

 writers who employ coordinates. Yet it is not necessary in the 

 analysis of Part II. 



§ 5. 



In fig. 28 the " lines of force " are drawn perpendicular to the in- 

 28 . difference loci. The directions of these lines of 



force are alone used in the formulae in Ch. II, § 9 

 which determine equilibrium. Therefore the 

 directions 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 one 

 point compared with another. The ratios of the 

 components at any point are important but these 

 ratios are the same whatever the length of the 

 arrow. Thus we may dispense with the total 

 utility density and conceive the " economic world " to be filled 

 merely with lines of force or ^' maximum directions." 



Even if we should give exact meanings to the length of these ar- 

 rows (so that the equation vUj = F(I) should signify not only that 

 for each position in the economic world a definite " maximum direc- 

 tion " exists but also that the rate of increase of utility or the length 

 of the vector along this line is given) — even then there would not be 

 a complete primitive U,= <?>(I) unless certain conditions were ful- 

 filled.* These conditions are (1) that the lines of force are so ar- 

 ranged that loci (surfaces in two dimensions, m — \ spaces in m di- 

 mensions) perpendicular to them can be constructed, and (2) that 



* Osborne, DifEerential Equations, p. 12. 



