90 
ON A SOLUBLE QUINTIC EQUATION. 
[914 
If, then, x 5 + qx 3 + rx 2 + sx + t = 0, we have the rational term = 0, and the 
coefficients of a, ¡3, y, S each = 0; in the class of equations under consideration, 
these last equations differ only in the signs of the radicals contained therein, so 
that one of them being satisfied identically, the others will be also satisfied. In 
particular, if q = 0, then a8 + ¡3y = 0 : the rational term gives 
A+B+G + D-10a8(A' + D'-B' -G') + t = 0, 
and the term in a gives 
5A" + 155" + 100" - 105" + ^ (G' 2 + 2D' 2 ) + ^ (O' + 2D') - 20a 2 8 2 + s = 0. 
For the equation x 5 + 3000« 2 + 20000« - 100000 = 0, the expression for the root is 
x=tyA + f/B-\-£/G + f/D, where 
A = 39000 + 18200 Vo + ( 1720 + 920 V5) V235 + 94 V5, 
5 = 39000 + 18200 V5 + (- 1720 - 920 V5) V235 + 94 V5, 
B = 39000 - 18200 V5 + (- 1720 + 920 J5) V235 - 94 V5, 
C = 39000 - 18200 V5 + ( 1720 - 920 Vo) V235 - 94 V5, 
and where also 
A' =- 
150- 
70 V5 + (- 
10- 
2 V5) V235 + 94 Vo, 
5' =- 
150- 
70 V5 + ( 
10 + 
2 V5) V235 + 94 Vo, 
B' =- 
150 + 
70 V5 + ( 
10- 
2 Vo) V235 - 94 V5, 
C =- 
150 + 
70 V5 + (- 
10 + 
2 V5) V235 - 94 V5, 
and 
4" = - 
940- 
100 Vo + (- 
100 + 
20 V5) V235 + 94 Vo, 
D" = — 
940 - 
100 V5 + ( 
100- 
20 V5) V235 + 94 Vo, 
5" =- 
940 + 
100 V5 + ( 
100 + 
20 V5) V235 - 94 V5, 
(7" = - 
940 + 
100 V5 + (- 
100- 
20 Vo) V235 - 94 V5. 
The foregoing forms 
are in 
some respects the 
most convenient; but it 
observed that we have 
A = 2600 V5 (1 + V5) ( 2 + Vo) + 40 (1 + V5) (18 + 5 y/5) Vl7 V5 (2 + Vo), &c., 
A' =- 10Vo(l + V5)( 2+ V5) - 2V5 ( 1+ Vo) V47 V5 (2 + V5), &c., 
A" = 20(1-V5)(18 + 13Vo) +20 Vo ( 1- Vo)V47 V5 (2 + V5), &c., 
or putting for shortness 
VQ = V47 V5 (2 + Vo), VQi = V- 47 V5 (2 - Vo),