120 OF REDUCTION. [CHAP. VIII. 



. C' 2 . _ 

 -yz+-yz + -y 



the interpretation of which is, Wherever the property A is present, 

 there either C is present and B absent, or C is absent. And in- 

 versely, Wherever the property C is present, and the property 13 

 absent, there the property A is present. 



These results may be much more readily obtained by the 

 method next to be explained. It is, however, satisfactory to 

 possess different modes, serving for mutual verification, of ar- 

 riving at the same conclusion. 



4. We proceed to the second method. 



PROPOSITION II. 



If any equations, V l = 0, F 2 = 0, fyc., are such that the develop- 

 ments of their first members consist only of constituents with positive 

 coefficients, those equations may be combined together into a single 

 equivalent equation by addition. 



For, as before, let At represent any term in the development 

 of the function F 1} Bt the corresponding term in the develop- 

 ment of F 2 , and so on. Then will the corresponding term in the 

 development of the equation 



Fx+Fo-f&c. = 0, (1) 



formed by the addition of the several given equations, be 

 (A + B + &c.) t. 



But as by hypothesis the coefficients A, B, &c. are none of them 

 negative, the aggregate coefficient A + B, &c. in the derived 

 equation will only vanish when the separate coefficients A, B, &c. 

 vanish together. Hence the same constituents will appear in the 

 development of the equation (1) as in the several equations 

 V l = 0, F 2 = 0, &c. of the original system taken collectively, and 

 therefore the interpretation of the equation ( 1 ) will be equiva- 



