86 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
By §8, because 7; is in a simple state, and (71) = 0, the coefficients 
1 
hy, e1, etc., are zero separately. But h; is clear of the surd eh Aig. 
therefore does not involve more than » — 1 distinct surds that are 
not roots of unity. Therefore, on the assumption on which we are 
proceeding, because hj = 0,42 = 0. In like manner, ¢, = 0, and 
soon. Therefore F' (72) = 0. 
§15. Cor. Let the simplified expression 7; be the root of an 
equation / (x) = 0 whose coefficients involve certain surds 
1 1 
nv $s . . . 
z, ,w, , etc. that have the same determinate values in 7; as in 
1 
F (x). Then, if 72 be a particular cognate form of # in which the 
1 1 
nh $ . . . 
surds 2, ,%, , etc, retain the determinate values belonging to them 
in 71, r2is a root of the equation F(x) = 0. For, F (r;1) = 0. 
Therefore, by the Proposition, F(#) = 9. Let R, restricted by the 
1 1 
condition that the surds z , Uy ae etc., retain the determinate values 
belonging to them in 71, be #’. Then #(#’) = 0. A particular case 
of this is /(r2) = 0. The corollary established simply means that 
1 1 
Te, 
the surds Z ,u, » etc., may be taken to be rational for the purpose 
in hand. 
§16. The simplified expression 7; being one of the particular 
cognate forms of A, let T1, Ta; etc. (5) 
be the entire series of the particular cognate forms of &, not 
necessarily unequal to one another. Then, if the equation whose 
roots are the terms in (5) be Y = 0, X is rational. In like manner, 
if those particular cognate forms of #, not necessarily unequal, that 
i 1 
5 S n s - E 
are obtained when certain surds % ,u, » ete, retain the determin- 
ate values belonging to them in 7, be 
71, Tey ete. (6) 
and if the equation whose roots are the terms in (6) be XY’ = 0, X’ 
1 1 
. ° ° % s . ° 
involves only surds found in the series 2, , u, » ete. This is sub- 
stantially proved by Legendre in his Théorie des Nombres, $487, third 
edition. 
