CHAP. VIII.] OF REDUCTION. 119 



and in these expressions replace, for simplicity, 

 1 - x by x, 1-ybyy, &c., 

 we shall have from the three last equations, 



xy (wz + wz) = 0;' (1) 



yz (xw + xw) = Q' } (2) 



xy=Hz; (3) 



and from this system we must eliminate w. 



Multiplying the second of the above equations by c, and the 

 third by c', and adding the results to the first, we have 



xy (wz + wz) + cyz (xw + xw) + c' (Icy - wx) = 0. 



When w is made equal to 1 , and therefore w to 0, the first mem- 

 ber of the above equation becomes 



xyz + cxyz + c'xy. 



And when in the same member w is made and w = 1, it be- 

 comes 



xyz + cxyz + c'xy - cz. 



Hence the result of the elimination of w may be expressed in the 

 form 



(xyz + cxyz + c'xy) (xyz + cxyz + c'xy - cz) = ; (4) 



and from this equation x is to be determined. 



Were we now to proceed as in former instances, we should 

 multiply together the factors in the first member of the above 

 equation ; but it may be well to show that such a course is not 

 at all necessary. Let us 'develop the first member of (4) with 

 reference to #, the symbol whose expression is sought, we find 



yz (yz + cyz - cz) x + (cyz + c'y) (c'y - c'z) (1 - x) = ; 

 or, cyzx + (cyz + c'y) (c'y - cz) (1 - x) = ; 



whence we find, 



(cyz + c'y) (c'y - c'z) 

 (cyz + c'y) (c'y - c'z) - cyz ' 



and developing the second member with respect to y and z 9 



