[311 
3als are 
such 
sidering it as 
on ; and in 
bions of 
the 
five values 
five several 
ratic radicals 
quintic ; and 
311] TO EQUATIONS OF THE FIFTH ORDER. 57 
corresponding to the systems 
a, /3, y, 8, 
/3, y, 8, cl, 
7, 8, cl, ¡3, 
a , & 7, 8, 
of the roots a, (3, 7, 8 ; i.e. the effect of the alteration of the values of the quadratic 
radicals is merely to cyclically permute the roots a, ¡3, 7, 8; and observing that any 
such cyclical permutation 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 x. 
The quantities cl, /3, 7, 8 are the roots of a rational quartic. If, solving the 
quartic by Euler’s method, we write 
cl = m + ^ F + V 6r+ H, VFGH = v, a rational function, 
/3 = m — \/~F+ 
7 = m + *JF- VG - Vi/, 
8 = m-\fF-\/G + \?H, 
then the expressions for F, G, H in terms of the roots are 
(cl + 7 — ¡3 — 8)\ (a + /3 — 7 — 8) 2 , (cl + 8 — ¡3 - y) 2 , 
which are the roots of a cubic equation 
tv’ - \v? + fiu — v- = 0, 
where A,, v are given rational functions of the coefficients of the quartic. We have 
\/G + \/H=J(\/G + VH)- = Jg + H + 2 VGH = /sJ\-F+y\/F- 
so that, taking © = F, the last-mentioned expressions for a, /3, 7, 8 will be of the 
assumed form 
The equation 
thus becomes 
that is, 
a. = m + V© + Jp + q \/©, &c. 
^©Vp + g'V© Jp — q V© = s 
^F\/(G — H)- = s, or F(G-Hf = s*; 
- F* + F(F- + G 2 + H 2 ) - 2FGH = s 2 ; 
or, what is the same thing, and putting © for F, 
C. V. 
— A© 2 + (A, 2 — p) © — 3^ = s 2 . 
8