into Conjugate Factors, 279 



subserve the purpose of leading to general theorems. In the 

 present communication, I shall confine myself entirely to the 

 expression (A), which I propose to employ for the purpose 

 of obtaining, in a more direct and simple manner, the formula 

 investigated in the last Number of this Journal (p. 192), and 

 of giving to that formula a greater degree of generality. 

 Let (f{x) be any function of a;; then since 



<Pi^)=f'<P{^h=fi'f'f{^)^> &c. 



we obviously have these identities, in which F, /, &c, are any 

 functions whatever : 



^(a;) = [F+v/{F2-^(a;)}]x[F-v'{F-<p(^)}] . . (A) 



= [F+v'{F2-/.^(a:)}] X [F-V{V'-f.f{x)}]^ 



= [F+^{F2-/.fW}] X [F-^{F2-/.f(^)}] 



X [f +^-[p2-i|] X [f'-v^J^F'2-^|] 



and so on. 



In the expression (A), let 



then, as F is entirely arbitrary, it may be made to take such 

 a value as to render the sum of the two conjugate factors in 



(A), that is to say 2F, equal to —p ; which value isF=— -p: 



and as the product of the same factors is q, it follows that these 

 factors are the roots of the quadratic equation 



x^+px + q = 0; 

 that is, the roots are 



4^V{7-*} (') 



Again : let there be the cubic equation 

 x^ + p.r^ -i-qx + rz^O, 

 and, in this case, put 



<p{x)=x^+j}x + q=-^; 



then, if a\ be one of the roots of the cubic, the expression 

 Xi^+pa?i + q 



will be the product of the two remaining roots ; and therefore, 

 that these roots may be represented by the factors in (A), we 



