B20 
ME. H. J. S. SMITH ON SYSTEMS OE LINEAE 
From the definition of as a greatest common divisor, which we have now obtained, 
we infer that if ||Dj| be any matrix containing another matrix ||v||, and if Dj, D^_i . . . Vs? 
Vs-i, ... be the greatest common divisors of the corresponding minors in ||D|| and ||v{| 
respectively, not only is Vs divisible by and Vs-i by Ds_i, but also by 
It is not difficult to show that in any matrix is the greatest common dmsor of 
all the quotients obtained by dividing each minor of order s by the greatest common 
divisor of its minors of order s—k. But as this extension of the preceding result is not 
needed in what follows, we may omit it here. 
We may add, that the theorem of this article is precisely equivalent to the following, 
which may be demonstrated by a different method. 
“ If P’* be the highest power of a given prime that divides all the minors of order s 
in a given matrix, and if all the minors of order s — 1 contained in one particular minor 
of order s are divisible by that minor is itself divisible by 
It should be observed that whenever all the minors of any determinant are zero, the 
quotient obtained by dividing the determinant by the greatest common divisor of its 
minors is also zero. 
Art. 17. These results admit of immediate application to the theory of systems of 
linear congruences. The general type of such systems is 
A,, , a;, + A,. 2 . . . A,. „ mod. M | 
^=1, 2, 3, ...^' J’ ^ 
and to construct a complete theory of them it is requisite, first, to assign a criterion for 
then’ resolubility or irresolubility ; secondly, when they are resoluble, to investigate the 
number of incongruous solutions of which they are susceptible ; and, lastly, to exhibit a 
method for obtaining all these solutions. We shall first suppose that n'=n ; i. e. that 
the proposed system is neither defective nor redundant. 
Let D„, D„_i .... v «5 V«-i? • • • respectively denote the greatest common divisors of 
the determinants and minors of the augmented and unaugmented matrices of the 
system (82.); also let S„_i, 
denote the greatest common divisors of M with 
v« 
V«-i’ 
of Mwith . . ., and let d„_i . . . similarly represent the greatest common divisors of M 
V»— 2 
Avith of M with 2^', &c. ; then, if d=d„ X d„_^ X X <^i, X X . • • X 
-L'n— 1 -L'n~2 
Ave have the two following theorems : 
(i.) “The necessary and sufficient condition for the resolubility of the system (81.) is 
d=ir 
(ii.) “ When this condition is satisfied, the number of its incongruous solutions is d." 
To demonstrate the first of these theorems, we revert to the principle of art. II, 
from which it appears that the necessary and sufficient condition for the resolubility of 
the .system (82.) is that the greatest divisors of the two matrices 
