RESOLUTION OF ALGEBRAICAL EQUATIONS. 1383 
the terms, ¢:, ¢25...... ,gur, are all the unequal cognate functions 
of f (p), obtained by giving definite values to all the surds inf (p) 
which are present in the coefficients of the powers of « in F, and 
forming the cognate functions without reference to the surd character 
of the surds thus rendered definite: F being understood to be gene- 
rated directly by the multiplication together of the factors, F, (x), 
F; (x), &e., and to have the coefficients of the various powers of % 
arranged so as to satisfy the conditions of Def. 8. 
For, all the terms in the series, 
O1 5 2 5 aieinielatatalels 9 Dinky ssid stele DONC NbO dod Gen aco ood BOS (1) 
are (by hypothesis) unequal. Suppose, if possible, that the terms, 
Bin Py eas ssn) Chorney Tale BeNCIER eis culL sud: (2) 
which are the roots of the equation, F, («) = 0, are not all unequal. 
Then, Fy (x), having equal factors, has a measure, H, of less dimen- 
sions, as respects «, than F, («), and yet involving, in the coefficients 
of the powers of «x, merely such surds as occur in Fy (x). But the 
surds in F (x) are identical with those in F, (x). [For instance, let 
1 a 
F, (x) =(1 + Jp)’, and, Fy (x) =2(1 + Spy i where 2 is a third 
root of unity, distinct from unity. The presence of 2 in Fy (x) does 
not affect the surds in the expression]. Therefore the expression 
H, of Jess dimensions as respects * than Fj (#), involves in the co- 
efficients of the powers of £ merely such surds as appear in F, (x): 
which, [since F\ (x) is the product of the terms, *—9, ,. ...., x—%n , 
where $1, 92,.--- ¢n, are all the unequal cognate functions of f (p) 
obtained by assigning definite values to certain surds in f (p)], is 
(Cor. 4, Prop. VI.) impossible. Therefore all the terms in (2) are 
unequal. Next suppose, if possible, that some term in (2) is equal 
toatermin (1). Then F, (x) and F, (x) have a common measure ; 
and their H. C. M. involves only such surds as appear in F, (x) 
or F, (x); that is, only such as appear in F, (x): which, as above, 
is (Cor. 4, Prop. VI.) impossible, unless F; (~) and Fy (x) are iden- 
tical. Suppose then, if possible, that Fy (x) =F, (x). The coeffici- 
ents of like powers of + must be equal. Let the coefficient of a 
certain powers of « in F; («), arranged according to the powers of y,, 
(we choose a coefficient where y; occurs in some of its powers), and 
satisfying (as, by hypothesis, it does) the conditions of Def. 8, be , 
