CHAP. X.] CONDITIONS OF A PERFECT METHOD. 155 



the present class of inquiries the chief aim of improvement of me- 

 thod should be to facilitate, as far as is consistent with brevity, 

 the transformation of equations, so as to make the fundamental 

 condition above adverted to universal. 



In connexion with this subject the following Propositions are 

 deserving of attention. 



PROPOSITION II. 



If the first member of any equation V satisfy the condition 

 V(\ - V) = 0, and if the expression of any symbol t of that equa- 

 tion be determined as a developed function of the other symbols, the 



coefficients of the expansion can only assume the forms 1, 0, -, -. 



For if the equation be expanded with reference to t, we ob- 

 tain as the result, 



Et+E(\-), (1) 



E and E' being what V becomes when t is successively changed 

 therein into 1 and 0. Hence E and E will themselves satisfy 



the conditions 



E) = 0, (2) 



Now (1) gives 



the second member of which is to be expanded as a function of 

 the remaining symbols. It is evident that the only numerical 

 values which E and E' can receive in the calculation of the co- 

 efficients will be 1 and 0. The following cases alone can there- 

 fore arise : 



1st. E'=l, E=l, 



2nd. E' - 1, E = 0, then E , _ E = 1. 



77" 



3rd. E = 0, E = 1, then -= - ^ = 0. 



4th. 



Whence the truth of the Proposition is manifest. 



