142 METHODS OF ABBREVIATION. [CHAP. IX. 



of those constituents in the development of y, whose coefficients 

 in the development (x, y) are 0. Viewing these cases together, 

 we may establish the following Rule : 



9. To deduce from a logical equation the relation of any class 

 expressed by a given combination of the symbols x, y, fyc, to the 

 classes represented by any other symbols involved in the given 

 equation. 



RULE. Expand the given equation with reference to the sym- 

 bols x, y. Then form the equation 



Et + E'(\-t) = 0, 



in which E is the product of the coefficients of all those constituents 

 in the above development, whose coefficients in the expression of the 

 given class are 1, and E' the product of the coefficients of those con- 

 stituents of the development whose coefficients in the expression of the 

 given class are 0. The value oft deduced from the above equation 

 by solution and interpretation will be the expression required. 



NOTE. Although in the demonstration of this Rule V is sup- 

 posed to consist solely of positive terms, it may easily be shown that 

 this condition is unnecessary, and the Rule general, and that no pre- 

 paration of the given equation is really required. 



10. Ex. 3. The same definition of wealth being given as in 

 Example 2, required an expression for things transferable, but not 

 productive of pleasure, t(\ - p), in terms of the other elements 

 represented by w, s, and r. 



The equation 



w - sip - str (!-/>) = 0, 



gives, when squared, 



w + stp + str (1 - p) - 2wstp - 2wstr (1 - p) = ; 

 and developing the first member with respect to t and p, 



(w + s- %ws) tp + (w + sr- 2wsr) t (1 - p) + w (1 - t)p 



+ w(l-)(l-p) = 0. 



The coefficients of which it is best to exhibit as in the following 

 equation ; 



0. 



