156 CONDITIONS OF A PERFECT METHOD. [CHAP. X. 



6. It may be remarked that the forms 1, 0, and appear in 

 the solution of equations independently of any reference to the 

 condition V(l - V) = 0. But it is not so with the coefficient . 



The terms to which this coefficient is attached when the above 

 condition is satisfied may receive any other value except the 



three values 1, 0, and -, when that condition is not satisfied. It 



is permitted, and it would conduce to uniformity, to change any 

 coefficient of a development not presenting itself in any of the 



four forms referred to in this Proposition into -, regarding this 



as the symbol proper to indicate that the coefficient to which it is 

 attached should be equated to 0. This course I shall frequently 

 adopt. 



PROPOSITION III. 



7. The result of the elimination of any symbols x, y, fyc.from 

 an equation F=0, of which the first member identically satisfies 

 the law of duality, 



V(\ - V) = 0, 



may be obtained by developing the given equation with reference to 

 the other symbols, and equating to the sum of those constituents 

 whose coefficients in the expansion are equal to unity. 



Suppose that the given equation V= involves but three 

 symbols, #, y, and , of which x and y are to be eliminated. Let 

 the development of the equation, with respect to t, be 



4* + JS (1 - = 0, (1) 



A and B being free from the symbol t. 



By Chap. ix. Prop. 3, the result of the elimination of x and y 

 from the given equation will be of the form 



JEfr+ #(l-f) = 0, (2) 



in which E is the result obtained by eliminating the symbols x 

 and y from the equation A = 0, E' the result obtained by elimi- 

 nating from the equation B = 0. 



