[914 
N. 
pp. 53—58.] 
ith Commensurable 
-130, has given, in 
: equations (founded 
the solution of a 
lution of the form 
ions involving only 
simple form: thus 
+ V^)" (2 + V5), 
is the proper form 
where observe that 
(2 + V5)V5, viz. the 
y by a factor 2 + \/5 
i foregoing equation 
the general theory. 
914] 
ON A SOLUBLE QUINTIC EQUATION. 
89 
which determine A", B", C", D" in terms of A', B', G\ D', a8, Ay; and then 
, A'A" n B'B 
A= -&r B 
n _ C'G" n DP" 
a8 ’ a.8 /З7 
which give A, B, G, D. 
If now we assume cc = a + /3 + y + 8, and regard A, B, G, D, A', B', C, D', 
A", B", C", D", a.8, Ay each as a rational function, we may express x, x 2 , x z , x 5 each 
of them by means of rational functions or of rational functions multiplied into 
a, A, y, & respectively: thus, 
x — ot А А Ay "b 8 
x 2 = a? + A 2 A 7 2 + 8 2 
= a + /3 + 7 + 3, 
= A'A 5(8 G'a D'y 
Ay + ol8 a8 Ay ’ 
2B'y 2A'8 _ * ct 
+ ^Ау + ~о8~ + 2а8 + 2 ^ + Пс8~ + 
2D'a , 2 G'A 
Ay ’ 
+ 2 a/3 A 2ary + 2a8 + 2/3y + 2/38 + 278 
&c.; and we thus obtain 
x 5 + qx 3 + rx 2 + sx At 
= A+B+G + D 
A (20aS + ЗО/З7) (A' A D') A 30 («8 + 20Ay) (5' + C') 
+ 3q (A' + B'+G' + D') + 2r(oc8+Ay) + t 
f G' 2 5B"Ay 1ЛЛ „ , 10D' 2 
+ «1М' + 5 ж + - 5 р + 100 + 
+ 1 0«*S* + 20 B" + 20 D" + 30/3=7* + ЗОЛ" /3 7 + бОа/ЗуЗ 
/В" 3D" „ . \ /V 2JJ’\ 
+ q {^ + ns +3a& + ^v +r ws + ~^s) + 
a8 
(7' 2D / \ 
^ + ^ + 5 ^-W + f-‘ 
a8 
+ 10/3V + 205" + 20G" + 30a 2 8 2 + 30C" ~ + 60a/3 7 8 
Ay 
/5" SC" OQ . л /4' 2C"\ ) 
+ s fe + ^7 + /3,y+ 8 ) + ЧЭу + Эу) + 1 
4 Kr+1 £ +e iS w+ 
105[ 2 
Ay 
a8 
+ 10/3y + 20^" + 205" + 30a 2 S 2 + 305" ~ + 60a/3 7 S 
+ 2 
^L" 35" 
+ 
Ay Ay 
10A' 2 
a 8 
(ТУ 25' 
4- —¡z—b SAy + 6a8 + r -w- + -о- + s 
\Ay Ay 
8 |б5" + 5i T + 5 + 105" + 
( ab ab 
A 10a 2 8 2 + 
/G" ЗА" _ * N /5' 2A'\ ) 
+ 3 U + ^ +3aS+6 ^) +? 'U + Y + T 
+ 10a*S* + 200'" + 20.4" + 30,OV* + 30,1" A + ОО3/З7О 
ab 
C. XIII. 
12