116 OF REDUCTION. [CHAP. VIII. 



The development of the second equation gives 



aO-y)-yO-*) = 5 



whence, x (1 - y) = 0, y (1 - x) = 0. (4) 



The constituents whose coefficients do not vanish in both deve- 

 lopments are xy, x (1 - ?/), and (1 - x) y, and these would to- 

 gether give the system 



**/ = 0, *(!-*,) = 0, (1-aOy-O; (5) 



which is equivalent to the two systems given by the developments 

 separately, seeing that in those systems the equation x (1 - y) = 

 is repeated. Confining ourselves to the case of binary systems 

 of equations, it remains then to determine a single equation, 

 which on development shall yield the same constituents with 

 coefficients which do not vanish, as the given equations produce. 

 Now if we represent by 



V, = 0, F 2 = 0, 



the given equations, F, and F 2 being functions of the logical sym- 

 bols x 9 y, z 9 &c. ; then the single equation 



F I + F,-0, (6) 



c being an arbitrary constant quantity, will accomplish the re- 

 quired object. For let At represent any term in the full de- 

 velopment F! wherein t is a constituent and A its numerical 

 coefficient, and let Bt represent the corresponding term in the 

 full development of F 2 , then will the corresponding term in the 

 development of (6) be 



(A + cB) t. 



The coefficient of t vanishes if A and B both vanish, but not 

 otherwise. For if we assume that A and B do not both vanish, 

 and at the same tune make 



A + cB = 0, (7) 



the following cases alone can present themselves. 



1st. That A vanishes and B does not vanish. In this case 

 the above equation becomes 



