130 METHODS OF ABBREVIATION. [CHAP. IX. 



CHAPTER IX. 



ON CERTAIN METHODS OF ABBREVIATION. 



1 . ^T^HOUGH the three fundamental methods of development, 

 -*- elimination, and reduction, established and illustrated in 

 the previous chapters, are sufficient for all the practical ends of 

 Logic, yet there are certain cases in which they admit, and espe- 

 cially the method of elimination, of being simplified in an im- 

 portant degree ; and to these I wish to direct attention in the 

 present chapter. I shall first demonstrate some propositions in 

 which the principles of the above methods of abbreviation are 

 contained, and I shall afterwards apply them to particular ex- 

 amples. 



Let us designate as class terms any terms which satisfy the 

 fundamental law V (1 - V) = 0. Such terms will individually 

 be constituents; but, when occurring together, will not, as do 

 the terms of a development, necessarily involve the same symbols 

 in each. Thus ax + bxy + cyz may be described as an expression 

 consisting of three class terms, a?, xy> and yz> multiplied by the 

 coefficients a, b, c respectively. The principle applied in the two 

 following Propositions, and which, in some instances, greatly 

 abbreviates the process of elimination, is that of the rejection of 

 superfluous class terms; those being regarded as superfluous 

 which do not add to the constituents of the final result. 



PROPOSITION I. 



2. From any equation, V= 0, in which V consists of a series of 

 class terms having positive coefficients, we are permitted to reject any 

 term which contains another term as a factor, and to change every 

 positive coefficient to unity. 



For the significance of this series of positive terms depends 

 only upon the number and nature of the constituents of its final 

 expansion, i. e. of its expansion with reference to all the symbols 



