152 CONDITIONS OF A PERFECT METHOD. [CHAP. X. 



length, it is desirable to show how, in the only cases which do 

 practically offer themselves to our notice, this source of com- 

 plexity may be avoided. 



The great primary forms of equations have already been dis- 

 cussed in Chapter viu. They are 



Whenever the conditions X (1 - X) = 0, Y(l - Y) = 0, are 

 satisfied, we have seen that the two first of the above equations 

 conduct us to the forms 



X(1-Y) = 0, (1) 



X(l- Y) + Y(l -X) = 0; (2) 



and under the same circumstances it may be shown that the last 

 of them gives 



v{X(l-Y)+Y(l-X)}=0', (3) 



all which results obviously satisfy, in their first members, the 

 condition 



V(\ - V) = 0. 



Now as the above are the forms and conditions under which the 

 equations of a logical system properly expressed do actually pre- 

 sent themselves, it is always possible to reduce them by the 

 above method into subjection to the law required. Though, 

 however, the separate equations may thus satisfy the law, their 

 equivalent sum (VIII. 4) may not do so, and it remains to 

 show how upon it also the requisite condition may be imposed. 



Let us then represent the equation formed by adding the 

 several reduced equations of the system together, in the form 



v + v + v" 9 &c. = 0, (4) 



this equation being singly equivalent to the system from which 

 it was obtained. We suppose v, v' 9 u", &c. to be class terms 

 (IX. 1) satisfying the conditions 



v(l-v) = 0, v(l -v') = 0, &c. 

 Now the full interpretation of (4) would be found by deve- 



