OF THE HIGHER DEGREES, WITH APPLICATIONS. 93 
1 
one of the coefticients of Xy be selected in which z * ig present, this 
coefficient can be shown to be unequal to the corresponding coefficient 
I 
in X, in the same way in which the terms in (15) were proved to be 
all unequal. 
§26. Cor. 4. Any two of the terms in (16), as Xj; and X,, being 
selected, the equations Y; = 0 and X; = 0 have no root in common. 
For, suppose, if possible, that these equations have a root in common. 
tA 
Taking the forms of Xy and Xj in (17) and (18), since 7; is a root of 
the equation XY, = 0, 
ec—1 
r+ (bz, d + etc.) an + ete. = 0. (19) 
1 
. . : € . ee . 
All the surds in this equation except z, occurin7,. It is impossible . 
1 1 1 1 
c ° c 
that z, can occur in 7; ; for, z, 
: C cen 
occurs in 7 ; andz, = Oz, 
. 1 
. . Ore . . . . c 
4, being a primitive ct* root of unity ; but this equation, if both z 
1 
1 
and z, occurred in 7;, would be of the inadmissible type (38). 
1 
Since z, does not occur in rj , it is a principal (see §2) surd in (19). 
We Breit beretore, keeping in view that 7; is the expression (1) in 
. m . eee 
which J, is a principal surd, arrange (19) thus, 
pol m—1 e—1 a2 
ce (4, a 44 ‘ (pz, © + PR, pe ot ete.) 
m—2 e—1 e—2 
== ¥ (112, yee 1h, Buel etc.) + etc. = 0; (20) 
1 
c e . on 
where 71, 91, etc., are clear of Zz Then, w; being a primitive 
1 
m** root of unity such that, by changing 4 nt into the m* root of 4, 
1 
a m 
whose value is w,J4 as! becomes 72 , 
