[914 
914] ON A SOLUBLE QUINTIC EQUATION. 91 
*jyi =0, and the 
ider consideration, 
tained therein, so 
also satisfied. In 
(so that, according to a foregoing remark, we have (2 + V5) VQ = VQi), then we have 
H = 40 (1 + V5) (65 V5 ( 2 + V5) + (18 + 5 V5) VQ}, &c., 
ul' =-2V5(l + V5){ 5(2+ V5)+ VQ}, &c., 
= 20 (1 - V5) { 18 + 13 V5 + V5 VQ}, &c„ 
where observe that the term 2 + V5 is a factor of Q. 
2 S 2 + s = 0. 
Starting from the values of A', B', C', D', we have 
A' = - 2 V5 (1 + V5) {5 (2 + Vo) + VQ}, 
D' = - 2 Vo (1 + Vo) {5 (2 + V5) - VQ}, 
and therefore 
for the root is 
H'D' = 20 (1 + V5) 2 (2 + V5) {25 (2 + V5) - 47 V5}, 
where the last factor is 
/5, 
= 50 — 22 V5, = — 2 V5 (11 — 5 V5), = - V5 (1 - V5) 2 (2 - V5). 
Hence 
/5, 
A'D' = - V5.20 (- 4) 2 (- 1) = 320 V5, 
that is, 
/5, 
A'D' = (aS) 2 ¡3<y — 320 V5, and similarly B'C' = a.8 (/3y) 2 = — 320 V5, 
whence 
A 
a8 = — 4 V5, /3y = 4 V5, and aS + /3y = 0, as above. 
\/5, 
We have, moreover, 
H / = — 2 V5 (1 + V5) {o (2 + V5) + VQ }, 
Vo, 
B' = 2 V5(l- V5){5(2-V5)-VQi}, 
and thence 
V5, 
= 80 {- 25 - VQQj + 5 (2 - V5) VQ ~ 5 (2 + V5) VQi}, 
V5, 
= 80 {- 25 - 47 V5 +(5 (2 - V5) - 5) VQ}, 
that is, 
A'B'+I3'Y= 4V5{-25-47 V5 + 5 (1-V5)VQ} 
V5, 
= - 20(47 + 5V5) + 20V5(l-V5) VQ, = H"; 
Vo, 
and similarly we verify the values of 5", C" and D". 
V5, 
We have next 
A'A" = 160 V5 {(10 + 5 V5 + VQ) (18 + 13 V5 + V5 VQ)}, 
V5. 
or observing that Q V5 is = 235(2 + V5), the whole term in { } is 
; but it is to be 
= (505 + 220 V5) + (470 + 235 V5) + (18 +13 V5 + 25 + 10 V5) VQ, 
= 975 + 455 V5 + (43 + 23V5)VQ =: 65V5(7 + 3V5) + (43 + 23 V5) VQ j 
5(2 + V5), &e., 
or we have 
5 (2 +Vo), &c., 
A'A" = 160 V5 {65 V5 (7 + 3 V5) + (43 + 23 V5) VQ}, 
= 160 V5 (1 + V5) {65 V5 (2 + V5) + (18 + 5 V5) VQ}, 
5(2 + V5), &c., 
and consequently 
A'A" -f- ^Sy = 40(1 + V5) {65 V5(2 + V5) + (18 + 5 V5) VQ}, = -4 ; 
and similarly we verify the values of 5, G, D. 
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