IXDETEEMINATE EQUATIONS AND CONOEUENCES. 
323 
(89.) 
Ai't. 18*. Let the proposed system of congruences be the defective system 
n+m+n mod. IVI, 
^■=l, 2, 3, ... w, 
and let the notation of the last Article be retained. It is easily seen that the condition 
of resolubility of the system (89.) is, as before, 
But the number of its incongruous solutions, when that condition is satisfied, is not I, 
but 5xM”*. For we have seen that we can find a unit-matrix |jaj|, and a prime matrix 
ijA'jl of the type nx{n-\-ni), satisfjdng the equation 
Nlx||A|i= 
1 ^n—1 
v? 
Vi 
J ^n—3 
. . ? 
Vi 
Vo 
XllA'II; 
we may therefore replace the system (89.) by a system of the form 
mod. M, 
in which 
and 
Vn— » 
Uj- — t, I “f" 2 H“ j 
®i, 1 -^“^1, n + m + l~\~^i, 2 -^2, n + m+lH" ^i,n -^n, iH 
(90.) 
If the system (89.) is resoluble, the system (90.) will be so too, and will give d or ^ different 
systems of values for Uj, U 2 , ... U„, any one of which may be represented by the formula 
U,•=^q, mod. M, 1 
. "o ( 91 -) 
1=1, 2, 3,... wj 
Let us replace the modulus M by P, the highest power of one of its prime divisors. 
Since |[A'|j is a prime matrix, one at least of its determinants, for example, the deter- 
minant 2+A',_ , A 2_2 ... A'„ „, is prime to P. It will follow from this that, whatever values 
we attribute to j-n+i, .t„+ 2 i ••• each of the ^ systems represented by (91.) is resoluble 
for the modulus P, and gives, for any assumed values of a:„+ 2 , • only one set 
of values of .r,, a’ 2 , . .. x„. Each of those 5 systems admits, therefore, of P’" solutions for the 
modulus P, ^■. e. of M"* for the modulus M. The system (89.) will consequently admit 
of $ X M” solutions. 
Let us also consider the redundant system of congruences, 
A,. A,. 2 ^ 2 +----+A,. „^„=A.. mod. M, | 
? = 1, 2, 3, . .. n-\-m, J 
and let denote the greatest di\isor of its augmented matrix. Let jp represent a 
prime dhisor of M, and let be the highest powers of p, which divide M, D^,, Vs 
respectively. The condition of resolubility of art. 11, applied to the system (92.), con- 
sidered with respect to the modulus j/, becomes, after division by^^'”“'^®, 

[This article has been added since the paper was read. The theorems contained in it are supplementary 
to that of the preceding article. September 1861, H. J. S. S.] 
(93.) 
