CHAP. VIII.] OF REDUCTION. 115 



other and kindred subjects, in order that their full proportions 

 may be understood. 



This chapter will exhibit two distinct modes of reducing 

 systems of equations to equivalent single equations. The first 

 of these rests upon the employment of arbitrary constant multi- 

 pliers. It is a method sufficiently simple in theory, but it has the 

 inconvenience of rendering the subsequent processes of elimina- 

 tion and development, when they occur, somewhat tedious. It was, 

 however, the method of reduction first discovered, and partly on 

 this account, and partly on account of its simplicity, it has been 

 thought proper to retain it. The second method does not re- 

 quire the introduction of arbitrary constants, and is in nearly 

 all respects preferable to the preceding one. It will, therefore, 

 generally be adopted in the subsequent investigations of this 

 work. 



2. We proceed to the consideration of the first method. 



PROPOSITION I. 



Any system of logical equations may be reduced to a single equiva- 

 lent equation, by multiplying each equation after the first by a dis- 

 tinct arbitrary constant quantity, and adding all the results, including 

 the first equation, together. 



By Prop. 2, Chap, vi., the interpretation of any single 

 equation, f(x,y ..) = is obtained by equating to those con- 

 stituents of the development of the first member, whose co- 

 efficients do not vanish. And hence, if there be given two equa- 

 tions, f(x,y..) = 0, and F(x, y . .) = 0, their united import will be 

 contained in the system of results formed by equating to all 

 those constituents which thus present themselves in both, or in 

 either, of the given equations developed according to the Rule of 

 Chap. vi. Thus let it be supposed, that we have the two equations 



^-2* = 0, (1) 



* - y = 5 (2) 



The development of the first gives 



- xy - 2tf (1 - y) = ; 



whence, xy = 0, x (1 - y) - 0. (3) 



i 2 



