122 OF REDUCTION. [CHAP. VIII. 



does not need (as it would remain unaffected by) the process of 

 squaring. Such equations are, indeed, immediately developable 

 into a series of constituents, with coefficients equal to 1, Chap. v. 

 Prop. 4. 



PROPOSITION IV. 



6. Whenever the equations of a system have by the above pro- 

 cess of squaring, or by any other process, been reduced to a form 

 such that all the constituents exhibited in their development have 

 positive coefficients, any derived equations obtained by elimination 

 will possess the same character, and may be combined with the 

 other equations by addition. 



Suppose that we have to eliminate a symbol x from any 

 equation V = 0, which is such that none of the constituents, in 

 the full development of its first member, have negative coefficients. 

 That expansion may be written in the form 



! and F being each of the form 



in which ^ t z . . t n are constituents of the other symbols, and 

 a l 2 a n in each case positive or vanishing quantities. The re- 

 sult of elimination is 



F, F 2 = 0; 



and as the coefficients in V l and F 2 are none of them negative, 

 there can be no negative coefficients in the product V l F 2 . 

 Hence the equation V l F 2 = may be added to any other equa- 

 tion, the coefficients of whose constituents are positive, and the 

 resulting equation will combine the full significance of those 

 from which it was obtained. 



PROPOSITION V. 



7. To deduce from the previous Propositions a practical rule or 

 method for the reduction of systems of equations expressing propo- 

 sitions in Logic. 



We have by the previous investigations established the fol- 

 lowing points, viz. : 



