CHAP. VII.] OF ELIMINATION. HI 



If from the last result we determine w 9 we have 







w 



l-s ' 



whence " All wealth is limited in supply." As p does not enter 

 into the equation, it is evident that the above is true, irrespec- 

 tively of any relation which the elements of the conclusion bear 

 to the quality "productive of pleasure." 



Resuming the original equation, let it be required to elimi- 

 nate s and t. We have 



w = st (p + r - pr). 



Instead, however, of separately eliminating s and t according to 

 the Rule, it will suffice to treat st as a single symbol, seeing that 

 it satisfies the fundamental law of the symbols by the equation 



st (1 - st) = 0. 

 Placing, therefore, the given equation under the form 



w - st (p + r - pr) - ; 



and making st successively equal to 1 and to 0, and taking the 

 product of the results, we have 



(w - p - r + pr) w = 0, 

 or w - wp - wr + wpr = 0, 



for the result sought. 



As a particular illustration, let it be required to deduce an 

 expression for " things productive of pleasure" (p), in terms of 

 "wealth" (w), and " things preventive of pain" (r). 



We have, on solving the equation, 



_w(l -r) 

 P ~w(\ -r) 



= ^wr + w(l - r) + 5(1 - w) r + ? (1 - w) (1 - r) 



= w(l -r) + -wr + -(l -w). 

 Whence the following conclusion: Things productive of plea- 



