855] 
419 
solution of (ci, b, c, d) = (a 2 , b\ c 2 , d 2 ). 
If p 2 —1=0, then either p = 1, giving q = 1, r = 1, and hence the equation is 
a? 4 +« 3 + « 2 + « + 1 = 0 ; or else jp = — 1, giving g = 0, r = — 1, and hence the equation is 
a? 4 — a? — x + 1 = 0, that is, (x — l) 2 (x 2 + x + 1) = 0. 
If p 2 + 2p — 8 = 0, then either p = 2, giving q = 3, r — 2, and hence the equation is 
x 4 + 2« 3 + 3a? 2 + 2a? + 1, that is, 
(a? 2 + « + 1) 2 = 0; 
or else p = — 4, giving ^ = 6, r = — 4, and hence the equation is x 4 — 4a? 3 + 6a? 2 — 4a? + 1 = 0, 
that is, (a? — 1) 4 =0. 
Secondly, if r= — l—p; here 
2q=p‘ 2 +p, 2 (—jJ 2 —p - 1) = q 2 — q ) 
the last equation multiplied by 4 gives 
8 (- p 2 —p — 1) = (p 2 + p) (p 2 + p — 2), 
that is, 
p 4 + 2p 3 + 7p 2 + 6p + 8 = 0, or (p 2 +p + 4) (p 2 + p + 2) = 0. 
If £> 2 + jp + 4 = 0, then p = {- 1 + i V(15)}, whence 
r = i{-l ±»V(15)}, 2q — p 2 +p, = — 4, or g = — 2 ; 
and the equation is 
{— 1 +i\/(15)} a? 3 — 2a? 2 + £ {— 1 + i\/(lo)} «+1=0. 
If p 2 +p + 2 = 0, then p = £ {— 1 ± i V(7)}; whence 
r = £ {-1 ± »V(7)}, 2q=p 2 +p, = - 2, or g = - 1; 
and the equation is 
a^ + H-l ± tVOOl^-^ + ii-l ±iV(7)}a? + l=0, 
that is, 
(a? — 1) [« 3 + {1 + i V(7)} a? 2 + £- {- 1 + i a/(7)} « — 1] = 0. 
We thus see that the eight equations are 
1 (a? — 1) 4 = 0, 
1 (a? 2 + x + l) 2 = 0, 
1 (a? — l) 2 (a? 2 + a? + 1) = 0, 
1 a? 4 + a? 3 + a? 2 + a?+l=0, 
2 («-1){« 3 + 1(1 ±^V7)a; 2 + ^(-l ±id1)x- 1} =0, 
2 + £ (- 1 ± » V15) a? 3 - 2« 2 + 1 (— 1 + iV15) a? + 1 = 0, 
8 
53—2