138 METHODS OF ABBREVIATION. [CHAP. IX. 



Ex. 2. Let us resume the symbolical expression of the defi- 

 nition of wealth employed in Chap, vn., viz., 



w = st {p + r(l -p)}, 

 wherein, as before, 



w = wealth, 



s = things limited in supply, 



t = things transferable, 



p = things productive of pleasure, 



r things preventive of pain ; 



and suppose it required to determine hence the relation of things 

 transferable and productive of pleasure, to the other elements of 

 the definition, viz., wealth, things limited in supply, and things 

 preventive of pain. 



The expression for things transferable and productive of plea- 

 sure is tp. Let us represent this by a new symbol y. We have 

 then the equations 



w = st {p + r(l - p)}, 



y = t p> 



from which, if we eliminate t and p, we may determine y as a 

 function of w 9 s, and r. The result interpreted will give the re- 

 lation sought. 



Bringing the terms of these equations to the first side, we 



have 



w-stp- sir (l-p) = 0. f 



y-tp = 0. 



And adding the squares of these equations together, 

 w + stp + sir (1 -p) - 2wstp - Zwstr (1 - p) + y+tp- tytp = 0. (4) 



Developing the first member with respect to t and p, in order to 

 eliminate those symbols, we have 



(w + s - 2ws + 1 - y) tp -f (w + sr - 2wsr 4- y) t (1 - p) 



+ (w + y) (1 - t)p + (w + y)(l- t) (l-p); (5) 



and the result of the elimination of t and p will be obtained by 

 equating to the product of the four coefficients of 



tp* '0 -p\ (1 - P> and (i - (i - p)- 



