16 Irving Fisher — Matheraatical mvestigations 



Since ft and ;/ are infinitesimal it follows from the mere mathe- 

 matical principle of continuity that : 



lit. of /i 2 = ut. of J/ 2, 

 (B, C, totals), 



.-. by (2) ut. of loOth loaf = ut. of ;/ 2, 



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



Hence our definition becomes : 



ut. of 100th loaf y 



ut. of 150th loaf ;/ 2 



= 2. 



Likewise 



ut. of 100th loaf __ ^ __ ^ 



ut. of 150th loaf 6 2 

 etc., etc., 



all of which results harmonize. 



Since C is any arbitrary quantity it follows that the definition of 

 the above ratio is independent not only of the particular commodity 

 employed as a means of comparison but also of the total quantity of 

 that commodity. 



It is to be noted here that if the utility of one commodity were 

 dependent 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 ratio 

 of two infinitesimal increments of the same commodity/ res-pect\\e]Y 

 equal in utility to the two utilities whose ratio is required, provided 

 these increments are on the margin of equal finite quantities : 



In general symbols this becomes : 



ut. of dA 



, --rv = n : — it lit. or dA = ut. oi ndM 



ut. oi dB 



(M total), 



and ut. of f7B = ut. of (TM 



(M also total), 



where ?i is any finite number, positive or negative, whole or frac- 

 tional. 



This definition applies not only to infinitesimal utilities of the 

 same commodity (as of the 100th and 150th loaves of bread) but to 

 those of different commodities or services. 



* We shall afterward see how this affects our notions of utihty (Part II, 

 Ch. IV. 



