IXDETEEMIN'ATE EQUATIONS AND CONGRUENCES. 
325 
or, finally, the number of solutions of the proposed system, considered with respect to 
M as modulus, is d or 
xArt. 19. We shall terminate this paper with an elementary theorem relating to linear 
systems of equations, which admits of frequent application in other parts of the theory 
of numbers. 
Resuming the notation of art. 11, we may see from the theorem of that article, that 
if the system (56.) be resoluble for any given values of the numbers Ai_o, •• • it 
is also resoluble for any other values of those numbers, respectively congruous, for the 
modulus D, to the given values ; so that the resolubility or irresolubility of the system 
depends exclusively on the residues of the numbers A, o, mod. D. There are D" possible 
combinations of these residues, and we shall now show that for D'‘~‘ of them the system 
is resoluble, while for the remaining D'‘~‘ (D — 1) it is irresoluble. For this purpose let 
|xiA|:=: 
D„ D„_, 
D, 
D„_i D„_2 Df 
XllA'il, 
(97.) 
||a|| denoting a unit-matrix, and ||A'|| a prime matrix of the same type as ||A||, while D„, 
I^n-15 Du Dq are of course the greatest common divisors of the determinants and minors 
of j[A|j. Let also 
— t^i=A,_o i+Ao , 0 a,, 2 +. ••+A„ 0 a,- 
The given system is then exactly equivalent to the system 
1’ ‘L “b A; 2 a’2+- • • + A,_ „ 
? = 1, 2, 3, ...??. 
4- m rt -b m 
J' 
(98.) 
D„_i- 
For the resolubility of this system it is requisite that C,- should be divisible by 
-L'n— t ^ 
and this condition is sufficient as well as necessary, because ||xA|| is a prime matrix. Now 
of the D or D„ values, incongruous for the modulus D, which may be attributed to C,-, 
-g — — are divisible by i whence it is evident that of the D" systems of values which 
may be attributed to C,, C’a, ... C„, D"^ system 
(98.) resoluble. Consequently the given system is also resoluble for D"~‘, and no more, 
of the systems of values that can be attributed (mod. D) to Ai_o, Aj^o, ... A„_o- 
Art. 20. The methods employed in the present paper are without exception such as to 
be immediately applicable to any species of complex numbers which admit of resolution 
into actual or ideal prime factors. And the greater part of the results at which we have 
arrived may be transferred, mutatis mutandis, to the theories of such numbers. For 
example, if in the equations (56.) we suppose the constituents of ||A|| to represent complex 
numbers, it will be found that the criterion for the resolubility or irresolubility of the 
system, which we have demonstrated in the case of ordinary integers, applies equally in 
the case of complex numbers ; and again, the condition of resolubility of a system of 
congruences of which the modulus as well as the coefficients are complex numbers, is 
.MDCCCLXI. 2 Y 
