OF THE HIGHER DEGREES, WITH APPLICATIONS. 117 
a ee 
mm m 
ey A 4 ie : 
Let # be the generic symbol under which the simplified expression 
e falls. By Prop. XVIII., when 4, is changed successively into 
the c terms in (54), e receives successively the determinate values 
61, €2, +--+», 3 and therefore e, 4; receives successively the 
determinate values 
@) 41, & 42, oa Camelens (60) 
There is therefore no particular cognate form of H4 that is not 
equal to a term in (60). By Prop. XVI. there is a value of the mt‘ 
root of unity ¢ for which the terms in (55) are severally equal, in 
1 
some order, to those in (39). Let the term in (39) to which ¢ 4 i 
n 
is equal be q; 4 ‘th Then, applying the principle of Cor. Prop. XV., 
as in Prop. XVI., it follows that the term in (39) to which 
m —2 M —2n 
m—2e, 4," in (55) is equal is 4,” 
and M— 2m being less than m. ‘Therefore ez 42 is equal to 
M 
, U being a multiple of m, 
2 “mm 
qi ky 4, 
respectively at equal distances from opposite extremities of the series. 
Bot 
. : ™ ™ . 
In other words, ¢: 42 is equal to an expression 0) ¢_,, in the 
second of the groups (59). In like manner every term in (60) is 
equal to an expression in the second of the groups (59). Let the 
unequal terms in (60) be 
, which is the product of two of the terms in (39) occuring 
€) 4, etc. (61) 
Then, by Prop. III., the terms in (61) are the roots of a rational 
irreducible equation, say fi (xv) = 0. Rejecting these, which are 
distinct terms in the second of the groups (59), it can in like manner 
be shown that certain other terms in that group are the roots of a 
rational irreducible equation, say fo (w) = 0. And so on. Ulti- 
mately, if f(a) be the continued product of the expressions /; (a), 
Jo (x), ete., the terms forming the second of the groups (59) are the 
roots of a rational equation of the (m — 1)t* degree. The proof 
applies substantially to each of the other groups. To prove the 
second part, it is only necessary to observe that, in the first of the 
groups (59), the last term is identical with the first, the last but one 
with the second, and so on. 
