of Algebraic Equation$ of the Fifth Degree. 563 



pose the function Yfuti) do not all of them belong to one 



group, but must have come in pairs from two of the three 

 groups (i.), (k.), (1.). Thus by the method of continuous sub- 

 stitutions we shall be conducted to an equation for Vp of the 

 form ^ ( V15 + C, V14 4- C^ V^-^ + . . + C.^f = 0, 

 in which the coefficients C,, Cg, • . Cjj will be rational func- 

 tions of Aj, Aj, . . Ag. And the roots of the equation V'^ 

 + C, V^"* + . . = will be expressible by 



= ^fu^y See theorem (f.) 



Again, observing that g^ —y% + j/y + "<i/a> ^"d therefore 

 &a.(^y) C^} ^ f^'a? ^^ ^"^^y ^s in the preceding case, 



Av(^.r)Cf) = (c^.)(r.O» 



and thence 



Now if we apply the interchange ("y s^ to the equations thus 

 obtained, reflecting that, since the index n may be suppressed, 

 there will result ^y'(yx)^{yi) = iV(y'0' ^ ^ff'' ^^ ^^'^^^ ^^^^® 



the last of which functions will manifestly be equal to 

 P/'C/s.) (^a/j^ ; since (^;^ and ^7^} are complementary rela- 

 tively to/,. 



That P///Sj\ and P^' do not belong to the same group, we 

 may at once convince ourselves from considering that 



Ha) = P" ^Xa:) = 1'- ^^{t) = '''■' 



where (^«) . ("^V^j). 



And in fact if we suppose /to be changed into /, 



^^^''OOn ^^'^ become P/'(45)^ 



and P„/ P^/ 



2P2 



