804 
MR. H. J. S. SMITH ON SYSTEMS OF LINEAR 
order, by the determinants ( — the results, we obtain 
{K h, ... K K,... 
f. c. since ^3, ... k„,) is not zero, 
(i, K h ,, . . . h„,_,)yi=0. 
The system (32.) is therefore equivalent to a system of m independent equations. 
Let j 7 ' represent the matrix of (33.), or of any other independent system 
equivalent to (32.) (the determinants of all such matrices are proportional); let 
r,. To, . . . be the determinants of Ijyl ; |j||| and E„ H2, . . . the matrix and deter- 
minants of the system similarly derived from (31.). By the theorem of art. 7, we have 
x|lj=2;H'x|4 
(37.) 
or observing that 
= 2.HX, and that (37) is an identity of the form 0=0, 
v.rCxly|=2PxlCi (38.) 
where jC| symbolizes the numbers Cj, Cg, ... C^, which correspond to the determinants of 
'/il in the same inverse order in which in equation (37.) the determinants of |)t|| corre- 
spond to those of [||||. But if 
of (32.) or (33.), we have also 
wx(w^+w) 
Q I 
represent a system of fundamental solutions 
^IxlyNSPxIfl, 
(39.) 
whence, combining (38.) and (39.), and representing the greatest common divisor of 
C„ C.,, . . . by c, Ave find 
CX|541C| (40.) 
If, then, ||cj| denote any square matrix of determinant c, and of the type 71 X n, the formula 
X li^li contains the complete solution of the problem. 
If 7 represent the greatest divisor of Ijylj, we infer from (38.) 
<7X|7| = 7X|C|, (41.) 
Avhence, if [jy'll be a prime matrix of the type mx(«i+w) satisfying the equation 
!7| = 7X|7'| (see art. 3), 
we find 
|C| = cx|7'| (42.) 
The preceding analysis enables us therefore to obtain simultaneously the representation 
of the determinants |C| as the determinants of two complementary matrices, of the types 
71 X (m+w) and m X {7n-\-n) respectively. We have thus two distinct methods of arriving 
at the solution of the problem, of which one requires the determination of a set of 
fundamental solutions of a system of linear equations ; the other the reduction (by the 
method of art. 3) of a given matrix to a prime matrix. The greatest divisor of lyjl, 
