148 METHODS OF ABBREVIATION. [CHAP. IX. 



wz + zw + ywz 

 ~ wz + zw + ywz-wz- ywz' 



wherefore, by development, 



x = Oyzw + yzw + yzw + Qy zw 

 + Qyzw + yzw + yzw + yz7>; 

 or, collecting the terms in vertical columns, 



the interpretation of which is 



In whatever substances the property A is found, there will also 

 be found either the property C or the property Z), but not both, or 

 else the properties B, C, and D, will all be wanting. And con- 

 versely, where either the property C or the property D is found 

 singly ', or the properties B, C, and D, are together missing, there 

 the property A will be found. 



It also appears that there is no independent relation among 

 the properties B, C, and D. 



Secondly, we are to find y. Now developing (5) with respect 

 to that symbol, 



(xwz + xwlz-\- xwz+ ~xzw)y + (xwz + xwz + Iczw + xzw) y = ; 

 whence, proceeding as before, 



y - xw'z + - (xwz + xwz + xzw), (7) 



0; (8) 



Q- 9 (9) 



0; (10) 



From (10) reduced by solution to the form 







r*g| 



we have the independent relation, If the property A is absent 

 and C present, D is present. Again, by addition and solution (8) 

 and (9) give 



__ 0_ 



xz + xz - w. 



Whence we have for the general solution and the remaining in- 

 dependent relation : 



