CHAP. XVIII.] ELEMENTARY ILLUSTRATIONS. 291 



* + (y + y*) * + ffi + ( z 



and making in the first member of this equation successively 

 y = 1, z = 1, y = 1, z = 0, &c., and multiplying together the 

 results, we have the expression 



(7 + t + u ) (? + if + ) (7 + t + u) (s + + M), 

 which is equivalent to 



(? + t + u) ( J -i- t -f w). 



This is the expression for J5J. We shall retain it in its present 

 form. It has already been shown by example (VIII. 3), that 

 the actual reduction of such expressions by multiplication, though 

 convenient, is not necessary. 



Again in (5), making x = 0, we have 



yzs + s (yz + y z + yz)*+ yi + ty + zu + uz = ; 



from which, by the same process of elimination, we find for E the 

 expression 



(?+ ~t\+ u) (s + ~t + u) (s + t + u) (s -f t + u) 



The final result of the elimination of y and z from (5) is there- 

 fore 



Whence we have 



_ _ (s + t+u) (s+t+u)(s+t+u)(s+t+u) 



or, developing the second member, 



+ - *stu + QTstu + Q~stu + 



Hence, passing from Logic to Algebra, 



stu + stu _ stu + ~stu _ stu + stu 

 -7- ,.'_ (7) 



= stu + stu +"stu + Htu + ~stu. 

 u 2 



