140 METHODS OF ABBREVIATION. [CHAP. IX. 



y = ws (1 - r) + Q wsr + - (1 - w) (I - s), (6) 



with w (1 - s) = 0. (7) 



The interpretation of which is 



1st. Things transferable and productive of pleasure consist of 

 all wealth (limited in supply and) not preventive of pain , an inde- 

 finite amount of wealth (limited in supply and) preventive of pain, 

 and an indefinite amount of what is not wealth jand not limited in 

 supply. 



2nd. All wealth is limited in supply. 



I have in the above solution written in parentheses that part 

 of the full description which is implied by the accompanying in- 

 dependent relation (7). 



8. The following problem is of a more general nature, and 

 will furnish an easy practical rule for problems such as the last. 



GENERAL PROBLEM. 



Given any equation connecting the symbols z,y..w 9 z.. 



Required to determine the logical expression of .any class ex- 

 ^essed in any way by the symbols x^ y . . in terms of the remaining 

 symbols, w, z, &c. 



Let us confine ourselves to the case in which there are but 

 two symbols, x, y, and two symbols, w, z, a case sufficient to de- 

 termine the general Rule. 



Let V - be the given equation, and let $ (x, y) represent 

 the class whose expression is to be determined. 



Assume t = (#, ^), then, from the above two equations, x 

 and y are to be eliminated. 



Now the equation V= may be expanded in the form 



Axy + Bx(\ -y)+C(\-x)y+D(l-x)(\-y) = 0, (1) 

 A 9 -B, C, and D being functions of the symbols w and z. 



Again, as (#, y) represents a class or collection of things, it 

 must consist of a constituent, or series of constituents, whose co- 

 efficients are 1. 



