220 EXAMPLE OF ANALYSIS. [c HAP. XIV. 



We shall on expression of the premises and elimination of the 

 indefinite class symbols (q), obtain the following system of equa- 

 tions : 



xyt 4 xyt = 0, 



tzw = 0, 



yv = 0, 



vx = 0, 



xz = 0; 



in which for brevity y stands for 1 - y, t for 1 - , and so on; whence, 

 also, 1 - f=t, 1 - y = y, &c. 



As the first members of these equations involve only positive 

 terms, we can form a single equation by adding them together 

 (VIII. Prop. 2), viz. : 



xyt + xyt + yv + vx + x z + tzw = 0, 



and it remains to reduce the first member so as to cause it to 

 satisfy the condition F (I - V) = 0. 



For this purpose we will first obtain its development with 

 reference to the symbols x and y. The result is 



(t + v + v + z + tzw) xy + (t + v + z + tzw) xy 

 + (v + tzw) xy + tzwxy = 0. 



And our object will be accomplished by reducing the four coeffi- 

 cients of the development to equivalent forms, themselves satis- 

 fying the condition required. 



Now the first coefficient is, since v + v = 1, 



1 + t + z + tzw 9 



which reduces to unity (IX. Prop. 1). 

 The second coefficient is 



t + v -4- z + tzw ; 

 and its reduced form (X. 3) is 



tvzw. 



The third coefficient, v + tzw 9 reduces by the same method 

 to v + tztvv; and the last coefficient tzw needs no reduction. 

 Hence the development becomes 



