232 
MR. G. T. BENRETT OR THE RESIDUES OF POWERS OF NUMBERS 
must lead to 
one value of (mod 2 ^^'), 
„ „ (mod 
&c. 
Substituting the values of the U’s in terms of the g's, 
{9{^'9t • • • 9iyy' {9i'^9t ■ ■ ■ • • • {9y'^9y'^- ■ • 9j^^Y'^ = 9y9r 
9i 
illXi + + • ’ • + CCfx ^ 22^2 
’ HfJ. ^'fj. 
9^ 
Vin + V 2'^2 + ... ^ 9^9^^ 
. (mod m), 
. g^!^ (mod m). 
Therefore (Proposition 7) it follows that 
hi^i + + . . . + = Ti (mod 
hi^v + ^ 22^3 + • • • + Vt. = 1-2 (mod p^^), 
Vi-^i + + . . . + (modj^jh), 
and we need that these shall have one set of values of (modulus X 2 (modulus 
p'^), &c., ... and only one. 
This being so the numbers U will generate completely the p)-power-exponeut 
numbers. 
We may suppose that the moduli . . . are in ascending order of magnitude, 
so that < ls + \- 
Suppose that 
— ^2 — ^3 — • • • — — h .+2 — • • • — h“^h+i — ^ 
6+2 — 
= lc<&c. ... = L 
Any one of the unknowns x, is to be found with regard to mod p^‘. 
We can write x^ in the form 
Obo 
= ^s + L.p^^ + + 
where 
£ < p'“ 
^sa < 
&c. 
This substitution gives in particular, 
(mod p''’). 
x^ = (mod p^“). 
x^ = 4 (mod i^"). 
Xa = (mod p^“). 
