OF THE HIGHER DEGREES, WITH APPLICATIONS. 83 
4 = 41, (3) 
where ¢; is an expression ‘involving no surds of so high a rank as 
1 
4, except such as either are roots of unity, or occur in 7; being at 
1 
the same time distinct from i . The expression 7; may then be 
said to have been simplified or to be in a simple state. 
$7. Some illustrations of the definition in §6 may be given. The 
root 83 cannot occur in a simplified expression +; ; for its value is 
2w, w being a third root of unity ; but the equation 84 = 2w is of 
the inadmissible type (3). Again, the root /5 cannot occur in a 
simplified expression ; for, ; being a primitive fifth root of unity, 
75 = 2 (w, + w4) + 1; an equation of the type (3). Once 
more, a root of the cubic equation x -- 32 — 4 = 0, in the form 
(2 + v3)t + (2 — vy 3)8, is not in a simple state, because 
(2 — y3)jt ~ (2— v3) (2 + v3). 
Bie isa 
§8. Let piAr is + pody “geen Des (4) 
‘ 
m., . : . . . 
where 4, isa surd occurring in a simplified expression 7; ; and pi, 
1 
2, ete., involve no surds of so higha rank as 4, ,except such as either 
are roots of unity, or occur in 7; being at the same time distinct 
1 
from 4," . The coefficients p1, ps, etc., must be zero separately. 
1 
For, by §5, if they were not, we should have wd” = l, w being an 
m* root of unity, and j involving only surds in (4) distinct from 
1 
aN 3 an equation of the inadmissible type (3). 
§9. The expression 7; being in a simple state, we may use # as a 
generic symbol to include the various particular expressions, say 
1, 72, T3, ete., obtained by assigning all their possible values to the 
surds involved in r;, with the restriction that, where the base of a 
surd is unity, the rational value of the surd is not to be taken into 
account. These particular expressions, not necessarily all unequal, 
may be called the particular cognate forms of R. For instance, if 
7, = 14, & has two particular cognate forms, the rational value of the 
