124 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
where L, LZ; , etc., are rational. Then, as above, Z; = 0. Keeping 
in view that » = 0, this means that m?q = 0. But gq is not zero, 
for this would make ,/ s = / 8; ; which, because we are reasoning 
on the hypothesis that ./ s; cannot have its value expressed in terms 
of ./ sand // z, is impossible. Therefore m is zero. And it was 
shown that » is zero. Therefore 4; = ./s, and 44=> — / 8s. 
Therefore 4; 43 = — / (p? — q? 2); which, because it has been 
proved that 4, 43 is not the root of a quadratic equation, is impossible. 
Hence 4/ s; cannot but be a rational function of / s and // z. 
$69. Proposition XXVI. The form of s is 
ALl+é)+Aryvil+ é), (72) 
h and e being rational, and 1 + e? being she value of z. 
By Prop. XXV., / s1:= v+ c/s, v and ¢ being rational 
functions of ./ z. Therefore s; = v? + c?s + 2ve./ 8s. By Prop. 
XXIV., ./ s is irreducible. Therefore ve = 0. Bute is not zero, 
for this would make ,/ s; = v, and thus / s; would be the root of a 
quadratic equation. Therefore v = 0, and / 3 =cf/s = 
(cq, + co 7 2) V 8, c and cg being rational. Therefore 
/(salHeV(P—-Pa=aAtavds (prayay» 
=(a4ptaqyat+Vz(agteap)=P+0V2 
Here, since p? — gz is rational, either ? = 0Oor @=0. As the 
latter of these alternatives would make ./ (p? — gz) rational, and 
therefore would make ./ (p +94 +/ 2) or / 8 reducible, it is inad- 
missible. Therefore c, p + cz qz = 0, and 
V(P-—Pa=(agt ap) Sz 
Now gz is not not zero, for this would make ,/ (ssi ) = + p; which, 
because ,/ s is irreducible, is impossible. ‘Therefore cg = 0. But, 
by hypothesis, ce, = 0; therefore ./s,, which is equal to 
(c, + c2 / 2) 8, is zero; which is impossible. Hence c; cannot 
be zero. We may therefore put ce = 1, and h (1 + e) = p. 
Thans=ptqV2z=A(1l+@)+AYU +e). Having 
obtained this form, we may consider z to be identical with 1 + é?, 
q with h, and p with 4 (1 + e? ). 
§70. The reasoning in the preceding section holds good whether 
the equation /’ (x) = 0 be of the first (see §30) or of the second 
class. If we had had to deal simply with equations of the first class, 
the proof given would have been unnecessary, so far as the form of z 
is concerned ; because, in that case, by Prop. VIII., 4; is a rational 
function of the primitive fifth root of unity. 
