OF THE FIFTH DEGREE. 129 
MmMU=J7tar/ 2%, and wuz = 9 —AV/%, (5) 
where g, a, are rational. It scarcely needs to be pointed out that 
these forms are valid whether the surd 4/ z is irreducible or not. 
Now Sz = 10 (uw uw + u2 3) = — 2 po. Therefore 
g = — Yo (P2). (6) 
Again, it was shown in the “ Principles” that the four expressions 
Ur ug, UZ U1, UZ U4, UZ U2, are the roots of a biquadratic equation. 
Boa by the same reasoning as that employed in the case of u, w, 
and uw, u,, the only surds that can appear in these expressions are 
J (he a ei J/2), J (he —h/z), and /z. Lethe + h/z = 8, and 
he —h/z=s,. Then 
va= (C— Via) Vaal yeVasleve 
Hence the expressions ue U3 , uz wy , uz Us, U2 uw, may have their 
values exhibited in terms of ,/ 2 and either of the surds 4/ s, \/ 5. 
Put 
Umuwy=kt+oYzt(OteoV2) Vs, 
Usug=ktoV/z—(0+oV/2)V8, (8) 
wm=k—ceVYzet+(0—e9V2V 5; 
Uwm=k—-cY2z—(0—-9V2)V 51; 
where k, c, 0, g, are rational. These coefficients must bear a relation 
to g, a, in (5). In fact, because 
(at uz ) (uj Uz) = (uy U4 )* (ue Uz ), 
(~@—@Wz\gta/2z2=(kKt+eV/2z’—(GO+¢V/2z) (lethy 2). 
Equating the rational parts to one another, and also the irrational parts, 
he(@+ ¢22+209) = Rh +P 2— 9 (9? — az, (9) 
h(@ + g*2 + 20¢2)= 2ke — a(g? — a 2). 
Because s2 = 15 13 ut u3)} = — 3ps, 
== "95 ( P3 )- (10) 
Tt will be convenient to retain the symbols g and k, whose values are 
(uw? us ) (uz ua )s 
U2 UZ 
given in (6) and (10). Again, because w? w2 = we 
have, from (5) and (8), 
