112 OF ELIMINATION. [CHAP. VII. 



sure are, all wealth not preventive of pain, an indefinite amount 

 of wealth that is preventive of pain, and an indefinite amount of 

 what is not wealth. 



From the same equation we get 



_! "(l~r)_ 

 /" w(l-r)~w(l-r) 9 



which developed, gives 



w(l-p) = -wr + -(l-w).r + -(l-w}.(\-r) 



Whence, Things not productive of pleasure are either wealth, pre- 

 ventive of pain, or what is not wealth. 



Equally easy would be the discussion of any similar case. 



17. In the last example of elimination, we have eliminated 

 the compound symbol st from the given equation, by treating it 

 as a single symbol. The same method is applicable to any com- 

 bination of symbols which satisfies the fundamental law of indi- 

 vidual symbols. Thus the expression p + r - pr will, on being 

 multiplied by itself, reproduce itself, so that if we represent 

 p + r-pr by a single symbol as ?/, we shall have the fundamen- 

 tal law obeyed, the equation 



y = y\ or y (1 - y) = 0, 



being satisfied. For the rule of elimination for symbols is founded 

 upon the supposition that each individual symbol is subject to 

 that law ; and hence the elimination of any function or combina- 

 tion of such symbols from an equation, may be effected by a sin- 

 gle operation, whenever that law is satisfied by the function. 



Though the forms of interpretation adopted in this and the 

 previous chapter show, perhaps better than any others, the di- 



rect significance of the symbols 1 and - , modes of expression 



more agreeable to those of common discourse may, with equal 

 truth and propriety, be employed. Thus the equation (9) may 

 be interpreted in the following manner : Wealth is either limited 

 in supply, transferable, and productive of pleasure, or limited in sup- 



