OF THE HIGHER DEGREES, WITH APPLICATIONS. 9I 
§23. Cor. 1. - Let iq +1 be any one of the particular cognate forms 
2 1 1 
of &; and let ae ha +1, ete., be respectively what A pAagreter. 
become in passing from 7, tov, 4 1. Also let the m particular cognate 
1 
m 
forms of F, obtained by changing 4 e 
in 1% + 1 Successively into the 
different m*” roots of dag 41, be 
Tatl,y Tat2s «+++ 39 Tatm-: (15). 
1 
These terms are all unequal. For, because 4, is a principal surd in 
1 
4 Mm a . 
7, and 72 1s What 7; becomes when 4 is changed into a surd whose 
1 
value is oJ,” , @, being a primitive m*® root of unity. the view may 
be taken that v2 involves no surds additional tu those found in 7, , 
except the primitive m' root of unity #,. Therefore ry — rz involves 
no surds distinct from primitive m roots of unity that are not found 
in the simplified expression 7}. Therefore 7; — rz is ina simple state. 
1 1 
m 
. : Tee 
Let ra +2 be what rq +1 becomes by changing 4 4.1 nto od | init 
Then 7¢4+1 — 7a +2 18 a particular cognate form of the generic 
expression under which the simplified expression ry — rg falls. 
Therefore 74 +1 — 1 +2 cannot be zero; for, if it were, 7; — 72 
would, by Cor. 2, Prop. ILi., be zero; which, by the proposition, is 
impossible. Hence, the first two terms in (15) are unequal. In like 
manner all the terms in (15) are unequal. 
§24. Cor. 2. Let X; = 0 be the equation whose roots are the 
terms in (14). When Xj is modified according to §21, it is, by §16, 
1 
clear of the surd ae Should it involve any surds that are not 
1 
: c ; 
roots of unity, take 2, a surd of the highest rank not a root of 
1 
. . c . . . 
unity in 4y ; and, when z, is changed successively into the different 
ch roots of the determinate base 2: , Iet 
i t ” ie 1) 
Nes Ce A 5 +. « amet ’ (16) 
y 
be respectively what X, becomes. Any term in (16), as 47, being 
selected, the m roots of the equation 1; = O are unequal particular 
