to Equations of the Fifth Order. 259 
a, B ‘? oD ) 
ends 8, @ 
Y> 6, a, B 
a, B ihe 6 
of the roots a, 8, y, 5; 7. e. the effect of the alteration of the 
values of the quadratic radicals is merely to cyclically permute 
the roots a, 8, y, 6; and observing that any such cyclical per- 
mutation gives rise to a like cyclical permutation of A, B, C, D, 
the alteration of the quadratic radicals produces no alteration in 
the expression for z. 
The quantities a, 8, y, 5 are the roots of a rational quartic. 
If, solving the quartic by Huler’s method, we write 
a=m+/F+/G+ /H, /FGH=y, a rational function, 
B=m—/F+/G—V/H, 
y=m+/F—/G—VH, 
S=m—/F —/G6+/H, 
then the expressions for F, G, H in terms of the roots are 
| (a+y—B—6)?, (a+B—y—8)*, («+6—B—y)’, 
which are the roots of a cubic equation 
u?—)u? + pu—v?=0, 
where A, “4, v are given rational functions of the coefficients of 
the quartic. We have 
JG4+V/H=V(VE+ caer erat om 
a OER Ree ~ VF. 
So that, takmg O=F, the last-mentioned expressions for «, 8,,6 
will be of the assumed form 
a=m+/O0+Vn+qV0, &e. 
The equation | 
/ OV p+qV OV p—qV O=s 
thus becomes 
Re /G Hi se or F(G—H)?=s?; 
that is, 
—F° + F(R’ + G?+ H*) —2EGH Ss? ; 
S2 
