in the theory of value and prices. 79 



could be found whatever the position of the plane. But the plane 

 must assume such an orientation that the center of gravity of these 

 points shall be the origin. That is the algebraic sum of all the A 

 coordinates consumed must equal the sum produced. Likewise the 

 algebraic sum of the B and C coordinates must each be zero. 



Hence with the geometrical analysis just described the equilib- 

 rium for a market of three commodities is determined when : 



(1) All individuals' combinations lie in a common plane through 

 the origin (each individual's sales and purchases cancel). 



(2) Each individual's combination is at the point where this plane 

 is tangent to an indifference surface for that individual (the point of 

 maximum net utility). 



(3) The points in the plane are so distributed as to make the origin 

 their centre of gravity (the production and consumption of each com- 

 modity balance). 



Whence it follows geometrically that the " maximum directions '^ 

 are parallel, their components (marginal utilities) proportional as 

 b^etween different individuals and that this proportion is that of the 

 orientation of the plane (the ratio of prices). 



When this equilibrium is attained, let us, through the j)oint of 

 tangency I, representing the consumption combination for I, pass a 

 section parallel to the plane of the A and B axes. The section of 

 this plane with the total income plane gives-a straight line which is 

 none other than the partial income line of Ch. I, § 11 and its section 

 with the indifference surfaces gives back the indifference curves of 

 Ch. I, § 10. 



§ 5. 



We have temporarily assumed only three commodities for we have 

 only three dimensions wherewith to represent them. A complete 

 presentation of the interdependence of utilities would require ni 

 dimensions, for the utility of any one commodity A, is subject to 

 m independent variations according to a change in any one of the 

 m commodities, though (in general) the change of the quantity A 

 itself is most important. 



There is a curious glamour over "the fourth dimension." The 

 popular interest is all to prove that it "exists." Its origin histor- 

 ically and its present usefulness is in the interpretation of a fourth 

 independent variation^ i. e. in representing just such relations as now 



