OF THE HIGHER DEGREES, WITH APPLICATIONS. 115 
In like manner each of the terms in (54) is equalto a term in (56). 
And, because the terms in (54) are unequal, they are severally equal 
to different terms in (56). By Prop. III., the terms in (54) are the 
roots of a rational irreducible equation, say ¢; (~) = 0, Rejecting 
from the series (56) the roots of the equation ¢; (x) = 0, certain of 
the remaining terms must in the same way be the roots of a rational 
irreducible equation ¢2 (x) = 0. Andsoon. Ultimately, if ¢ (x) 
be the continued product of the expressions 4; (x), ¢2 (a), ete., the 
terms in (56) are the roots of the rational equation g (a) = 0. 
§53. The equations ¢ (x) = 0, ¢2 (x) = 0, etc., formed by means 
of the expressions ¢; (x), 2 («), etc., may be said to be sub-auxiliary 
to the equation / (x) = 0. It will be observed that the sub- 
auxiliaries are all irreducible. 
§54. Proposition XVIII. In passing from r; to r,, while 4 
becomes 4,, the expressions a , 6; , which, by Prop. XIV., involve 
only surds occurring in 4; , must severally receive determinate values, 
Gn, bn, etc. In other words, a being a particular cognate form of 
A, there cannot, for the same value of 4, , be two particular cognate 
forms of A, as a, and ay , unequal to one another. And so in the 
case of 6; , e,, etc. 
For, just as each of the terms in.(42) is equal to a term ia (41), 
there are primitive m*® roots of unity t and 7’such that the expressions 
1 2 1 1 
T 4, + 7 adn 4 +ete, 7 Ae Ose T2 an A. + etce., 
are equal to one another. Therefore, if dy = 4,,, in which case, by 
1 
assigning suitable values to t and 7’, 4 na may be taken to be 
1 
mm 
equal to 4 , 
ee es 
A(t — 7) + 4." (a7? — ay T?) + ete. = 0. (57) 
Suppose if possible that the coefficients of the different powers 
1 1 
of 4,” in (57) are not all zero. Then, by §5, ¢ A = 1); ¢t being 
an m*‘ root of unity; and /, involving only surds of lower ranks 
1 1 
than 4," - Hence, by Prop. XV. and Cor. Prop. XV, 4 ri is a 
1 
rational function of surds of lower ranks than 4 x and of the 
