280 Prof. J. R. Young 07i the decomposition of Functions 

 shall merely have to determine F so as to fulfill the condition 



and, consequently, the two remaining roots are 



as in the paper referred to above 



Let us now take a biquadratic equation 



a/^ -\-px^ + qx^ -f r.ar + 5 = ; 

 then if x^ and Xc^ be two of its roots, we shall have 



x^ +pxc^ + ga^a + r = 



X<2 



.-. {x,^-:^i)+p{x,--xi)^.q{x,-x,)^ ^- " ^ = '"^^^^^ 

 and, consequently, the expression 



x^ H- x^x^ + x^ +p{xx + ^2) + 2' = = (^1 + -^2) 





1<*2 



(B) 



+p{xx + "^2) — ^i-^a + !Z 



is the product of the two remaining roots. 



Let (p[x) represent this product: then, as before, in order 

 that the two factors in (A) may represent these roots, we 

 shall merely have to determine F so as to fulfill the condition 



2F=-Gr,+.'2+i^).-. F=- ^i±|i±^; 



so that the expressions for the two remaining roots are 



__ Xy + x^+p 

 2 



the irrational part of which may be put in the more convenient 

 form 



