CHAP. X.] CONDITIONS OF A PERFECT METHOD. 153 



loping the first member with respect to all the elementary symbols 

 x, y, &c. which it contains, and equating to all the constituents 

 whose coefficients do not vanish ; in other words, all the consti- 

 tuents which are found in either v,, v', v", &c. But those consti- 

 tuents consist of 1st, such as are found in vj 2nd, such as are 

 not found in v, but are found in v' ; 3rd, such as are neither found 

 in v nor v' 9 but are found in v", and so on. Hence they will be 

 such as are found in the expression 



v + (l-v) v' + (1 - v) (1 - v) v" + &c., (5) 



an expression in which no constituents are repeated, and which 

 obviously satisfies the law F(l - V) = 0. 

 Thus if we had the expression 



(1 - t) + v + (1 - z) + tzw, 



in which the terms 1 - t, I - z are bracketed to indicate that they 

 are to be taken as single class terms, we should, in accordance 

 with (5), reduce it to an expression satisfying the condition 

 P^(l - V) = 0, by multiplying all the terms after the first by t, 

 then all after the second by 1 - v ; lastly, the term which remains 

 after the third by z ; the result being 



1 - t + tv + t(\ - v) (1 - z) + t (1 - v) zw. (6) 



4. All logical equations then are reducible to the form F= 0, 

 V satisfying the law of duality. But it would obviously be a 

 higher degree of perfection if equations always presented them- 

 selves in such a form, without preparation of any kind, and not 

 only exhibited this form in their original statement, but retained 

 it unimpaired after those additions which are necessary in order 

 to reduce systems of equations to single equivalent forms. That 

 they do not spontaneously present this feature is not properly 

 attributable to defect of method, but is a consequence of the fact 

 that our premises are not always complete, and accurate, and in- 

 dependent. They are not complete when they involve material 

 (as distinguished from formal) relations, which are not expressed. 

 They are not accurate when they imply relations which are not 

 intended. But setting aside these points, with which, in the 

 present instance, we are less concerned, let it be considered in 

 what sense they may fail of being independent. 



