16

Irving Fisher—Mathematical investigations

Since /? and y are infinitesimal it follows from the mere mathe-matical principle of continuity that:

ut. of /S/2 = ut. of yj 2,

(B, C, totals),

by (2) ut. of 150th loaf = ut. of yj 2,

(150 loaves, C, totals) Q. E D.

H tuice our definition becomes :

ut. of 100th loaf _ yut. of 150th loaf yj'l

Likewise :

ut. of 100th loaf _ 8 _

ut. of 150th loaf 8/2 ’

etc., etc.,

all of which results harmonize.

Since C is any arbitrary quantity it follows that the definition ofthe above ratio is independent not only of the particular commodityemployed as a means of comparison but also of the total quantity ofthat commodity.

It is to be noted here that if the utility of one commodity weredependent on the quantities of others, two applications of the defini-tion would yield discordant results.*

We may state our definition in general terms as follows :

(3) The ratio of two infinitesimal utilities is measured by the ratioof two infinitesimal increments of the same commodity respectivelyequal in utility to the two utilities whose ratio is required, providedthese increments are on the margin of equal finite quantities:

In general symbols this becomes :

ut. of dAUt. of dB

= n :—if ut. of dA = ut. of ndM(M total),

and ut. of dB = ut. of dM(M also total),

where n is any finite number, positive or negative, whole or frac-tional.

This definition applies' not only to infinitesimal utilities of thesame commodity (as of the 100th and 150th loaves of bread) but tothose of different commodities or services.

* We shall afterward see how this affects our notions of utility (Part II,Ch. IV.