92 PRINCIPLES OF THE SOLUTION OF EQUATIONS 
1 
c = *. ys 
cognate forms of #. For, z, beimg a c* root of ~ distinct from 
1 1 1 
Cc c Ig 
ae let rq +1 be what 7; becomes when z, becomes Za the ex- 
te 1 
pressions 4, , hy , etc., at the same time becoming 4) val ha +1, ete. 
Then we may put 
e—1 c—2 
X) = x™ + (bz,° + dz,° + etc.) a™—14 ete.; (17) 
1 
where 6, d, etc., are clear of z + . Therefore, because 7; is a root of 
the equation XY; = 0, 
1 m—t1 
et (hide etc.) |™ 
( m 1 
ec—1 c—2 m—1 
+ (bee da cof ete.) } — ~ (nd m + ete.) tat + ete. = 0. 
All the surds in this equation occur in the simplitied expression 7; . 
Therefore, by Prop. 1L., 
chy m—1 
=e ee 
atl 
e—1 m—1 
Je (bz 2 + dee c amt ete ) | = Eke +14 cay _™ + ete.) }™~ 7 pete. =0. 
m—t1 
Therefore; (Aa +14 Ee , + etc.) or 7g +1 is a root of the equation 
7 c—1]1 
Ay = a™ + (62, ° ae etc.) x*—1 +4 ete. = 0. (18) 
Therefore also, by Cor. Prop. IL., all the terms in (15) are roots of 
that equation. And, by Cor. 1, the terms in (15) are all unequal. 
Therefore the equation X; = 0 has m unequal particular cognate forms 
of & for its roots. 
, 
§25. Cor. 3. No two of the expressions in (16), as x; and Xj, are 
’ 
identical with one another. For, in order that X, and X, might be 
identical, the coefficients of the several powers of x in X, would need 
/ 
to be equal to those of the corresponding powers of in X; but, if 
