in the theory of value and prices.

91

In the fourth set of equations there are really

separate equations but only n\3m — 1) are independent. Which shallbe selected is a matter of convenience. We may make every equa-

tion contain p am for instance and write

P*,*'-Pi>,* =

d\J " d\JdA.dB„,

<ZU dU

d'K, 1

= etc.

P°.„ : ~ etc.

P*,n • Pa. K = ,

Pa,„--P>,'=,

Now from the first two equations'we may derive by division

Pt>,„ - Pc,

_cTU

d\JdC ’

TTfl

but we might wish to use this last as one of the n(3m—l) indepen-dent equations, if A, ilr should “fail.” From the n(3m —1) — sepa-

rate equations we are at liberty to select for use any n{3m — 1) inde-pendent ones; and if in this selection there occur any which by somechange of quantities fail, we are compelled to change our selection sothat the new n(3m —1) equations shall avoid the “failing” magni-tudes.

This is interpreted in the mechanism as follows : when a cisternis wholly above the surface of the tank (as IIIC fig. 8) and so con-tains nothing, the quantity of commodity and its utility “fail.” Thelevers which keep the ordinates in proportion to the correspondingordinates of other individuals may be far more numerous than thelevers pictured in former diagrams. Thus for four cisterns theremay be six levers (by joining each pair) but only three are neces-sary. The “ failure ” of any magnitudes will not invalidate anysystem of levers originally selected ; it will simply make their num-ber greater than necessary.

II. THE CISTERNS AND DIAGRAMS OF PART I COMPARED‘ WITH THE DIAGRAMS OF JEVONS AND OFATJSPITZ UND liIEBEN.

§ 1-

In order to represent geometrically the relations between quantityof commodity, marginal utility, total utility, and gain (any two ofwhich four magnitudes are determined by a specified relation between