CHAP. VIII.] OF REDUCTION. 117 



and requires that c = 0. But this contradicts the hypothesis that 

 c is an arbitrary constant. 



2nd. That B vanishes and A does not vanish. This assump- 

 tion reduces (7) to 



,1 = 0, 



by which the assumption is itself violated. 



3rd. That neither A nor B vanishes. The equation (7) then 

 gives 



- A 

 C = 1T' 



which is a definite value, and, therefore, conflicts with the hy- 

 pothesis that c is arbitrary. 



Hence the coefficient A + cB vanishes when A and B both 

 vanish, but not otherwise. Therefore, the same constituents 

 will appear in the development of (6), with coefficients which do 

 not vanish, as in the equations V l = 0, F 2 = 0, singly or together. 

 And the equation V i + c V 2 = 0, will be equivalent to the sys- 

 tem V l = 0, F 2 = 0. 



By similar reasoning it appears, that the general system of 



equations 



V, = 0, F 2 = 0, F 3 = 0, &c. ; 



may be replaced by the single equation 



F 1 + cF 2 + e'F 3 + &c. = 0, 



c, c', &c., being arbitrary constants. The equation thus formed 

 may be treated in all respects as the ordinary logical equations 

 of the previous chapters. The arbitrary constants c l5 <? 2 , &c., are 

 not logical symbols. They do not satisfy the law, 



c, (1 - Cl ) = 0, c, (1 - c 2 ) = 0. 



But then- introduction is justified by that general principle which 

 has been stated in (II. 15) and (V. 6), and exemplified in nearly 

 all our subsequent investigations, viz., that equations involving 

 the symbols of Logic may be treated in all respects as if those 

 symbols were symbols of quantity, subject to the special law 

 x (1 - x) = 0, until in the final stage of solution they assume a 

 form interpretable in that system of thought with which Logic 

 is conversant. 



