FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
233 
= ia+i + (mody^). 
Xa+2 = L+2 + L+2.aP^° (mod p^‘). 
Xi =^b-\- ^bap’“ (mod p^‘). 
= ^ 6+1 + ^ 5 + 1 ,..^“ + I 6 + 1 .& (mod/'). 
^6+2 = ^6+2 + ^b+2.aV'" + ^6 + 2.6 (mod /'). 
Xc =t-\- (mody'). 
&c. &c. 
First replace each modulus which exceeds by p^", and substitute the assumed 
values of the x’s. 
We get the /x congruences 
+ ho4 + . . . + = Ii (mod p^“), 
4ifi + 42^ + •••-!- = I 2 (mod 
4i^i + V 24 + . . . + = 1^. (mod p^“), 
and we need that these shall determine a single set of values for ^ 2 , . . . 
(mod p^“). 
The necessary and sufficient condition is that the determinant 
(hn 42> H?>'- • • • W) 
should be prime to p (Proposition 28). 
Suppose that this is so, and that the numbers fj, ^ 0 , . . . are determined. 
When these values are substituted above, suppose that the values of the left-hand 
sides are I 3 —&c. . . . (where the negative signs are written for 
convenience in what follows, and 1\, 1\, . . . are therefore negative). 
Next replace each modulus which exceeds p^'' by p^\ and substitute the assumed 
values of the as’s. 
The first a congruences are already satisfied. 
The rest give 
1« + 1 -/"Fa+l + P^'‘ (4 + l.„+l L^l.a + 4+l.a+2^a+2.« + • • • + i. + i.^ 4«) = ^a + l (mod^^). 
Ia+2 — y- 14+2 + y- {i.+2.a+l L+l.a + 4+2.a + 2 L+2.a + • • • + 4 + 2.;a = Ia + 2 (mod p^‘). 
IV + P^’ (V.«+I L.+ \.a + V« + 2 ^«+2.a H- . . . V = If* (mod 
MDCCCXCIIT.—A. 2 H 
