№ 
88 
[914 
914. 
ON A SOLUBLE QUINTIC EQUATION. 
[F rom the American Journal of Mathematics, vol. xm. (1891), pp. 53—58.] 
Mr Young, in his paper, “ Solvable Quintic Equations with Commensurable 
Coefficients,” American Journal of Mathematics, x. (1888), pp. 99—130, has given, in 
illustration of his general theory of the solution of soluble quintic equations (founded 
upon a short note by Abel), no less than twenty instances of the solution of a 
quintic equation with purely numerical coefficients, having a solution of the form 
fyA + f/B + f/C + y/B, where A, B, G, D are numerical expressions involving only 
square roots. But the solutions are not presented in their most simple form: thus 
in example 1, x 5 + 3« 2 + 2x — 1 = 0, the expression involves a radical 
(21125 +9439 fo): 
here 
(21125 + 9439) V5, = (9439 + 4225 fh), = V5 . £ (18 + 5 fof (1 + fof (2 + J5), 
so that, taking out the roots of the squared factors, we have as the proper form 
of the radical the very much more simple form V47 (2 + fo) \Jh; where observe that 
(2 + \/5) (2 — f5) = — 1, and thence (2 + V— 47 (2 — f5) = V47 (2 + f5) f5, viz. the 
conjugate radicals V— 47 (2 — f5) J5 and V47 (2 +f5)fo differ only by a factor 2 + f 5 
which is rational in 1 and fh. To avoid fractions I consider the foregoing equation 
under the form 
« 5 + 3000« 2 + 20000« -100000 = 0, 
and I will presently give the solution; but first I consider the general theory. 
Writing 
A'= a 2 y, 
B' = a/3 2 , 
C' = 7 % 
D'=/38 2 , 
Also 
A = a 5 , 
B = /3 5 , 
C = f, 
D= 8\ 
we have A'JD' = a?8 2 /3y, B'C' = a8/3 2 f 
A" = a?/3, 
B" = ¡3 S 8, 
C" = af, 
JD" = 7 S 3 , 
A" = 
A'B' 
B" = 
B'D’ 
a8 
G" = 
A'C' 
oi8 
ry, CU 
/37 
wh 
&c.