556 
rea D013) 
The number of the prime roots corresponding to all the different 
divisors of m is therefore 
1 a ip 1 B 
Re ie De 
Dd q/ 1 
TN A 
5 Eee Sese. .Ò =m. 
t 1 
Now these prime roots being all different, they must satisfy an 
equation of degree m, which, because every one of these roots is 
also a root of 1—«v" =O, must coincide with 1—a” — 0. 
To illustrate this theorem, put m= 20 — 2? . 5, then the divisors are 
Us A 45 Oo MO, ZO 
and the factors corresponding to the prime roots 
l—a, Ita. 14-27, 1ta4 27+ 2*. 1—a?— oaf, l—a?+ a4 ada! 
or 
l—a, 1—2?, lef, 1e), 1—2z, 1, 
The continued product of these factors is evidently 1—«?°, or 
6 
la? = (ledi). 
ii 
Developing in the same way the several factors of @ (x), we 
may write 
Ti To 2 7 
DOMUS US) re (ISEB Te 5 ar 0 (4) 
where the quantities «;, ranged according to ascending magnitude, 
represent the different divisors of a,,a,..a,, and 7; the numbers of 
the divisors «;. We may remark here that ¢,—=1 and r, =7. 
If, for instance 
a, +20e,+ 52, + 100, 4+ 202,=n 
is the given equation, we have 
® (x) = (le) (la?) (1 —#) (1 — 2°) (1— 2”) 
where 
La = lr 
Nay = Nang lee 
EE 
IPS tlw oo log? 5 lw 
lg a , la lea a Nee Og NG 
10 30 
5 ll 
hence 
@ (x) =(i—z)® (le) (et) (la) (lS =<")? ASN) 
