128 OF REDUCTION. [CHAP. VIII. 



Whence there will result 



stq (l-r) = 0, sir (1 - q) = 0, 



these being equations which are deducible from either of the 

 primitive ones. Their interpretations are 



Similar triangles which have their corresponding angles equal 

 have their corresponding sides proportional. 



Similar triangles which have their corresponding sides propor- 

 tional have their corresponding angles equal. 



And these conclusions are equally deducible from either pre- 

 miss singly. In this respect, according to the definitions laid 

 down, the premises are not independent. 



14. Let us, in conclusion, resume the problem discussed in 

 illustration of the first method of this chapter, and endeavour to 

 ascertain, by the present method, what may be concluded from 

 the presence of the property C, with reference to the properties 

 A and B. 



We found on eliminating the symbols v the following equa- 

 tions, viz. : 



xy(wz + wz) = Q, (1) 



yz (xw + xw) = 0, (2) 



Icy = wz. (3) 



From these we are to eliminate w and determine z. Now (1) 

 and (2) already satisfy the condition F(l - F) = 0. The third 

 equation gives, on bringing the terms to the first side, and 

 squaring 



xy (1 - ~cz) + w~z(l - xy) - 0. (4) 



Adding (1) (2) and (4) together, we have 



xy (wz + wz) + yz (xw + x) + xy(l-wz) + wz(l- xy) = 0. 

 Eliminating w, we get 



(xyz + yzx + wy) {xyz + yzx + xyz + z(l - xy)} = 0. 



Now, on multiplying the terms in the second factor by those in 

 the first successively, observing that 



xx = 0, yy = 0, z~z = 0, 



