m)ETEEMI]S^ATE EQUATIONS AND CONGEUENCES. 299 
can always be done, either by trial, or by other ob\ious and not unsynimetrical expe- 
dients. 
Art. 5. If j|fi![| represent the matrix of a complete set of independent solutions of the 
proposed system (16.), and |jJ|| be any matrix of the same type as ||«||, and connected with 
|«|| by the equation 
■FINISH x!l4 (22.) 
in which j|>^.’|j denotes a square matrix of which the determinant is not zero, it is evident 
that the constituents of jj5j[ are also a complete set of independent solutions. And, con- 
versely, if j|J[| be the matrix of a complete set of independent solutions, ||a|| is also the 
matrix of a similar set. For if ||K|| be the matrix composed of the first minors of P'jj, so 
that 
we have from ( 22 .), 
||K|ix|i^||=||/5:,yI, ... AUixH; 
from which it appears that . . . j|x|j«i|, and therefore ||a|| itself, is the matrix of an 
independent set of solutions. 
This observation enables us to obtain a complete set of relatively prime solutions, as 
soon as we have obtained an mdependent set. If ||^}| be the matrix of the independent 
set, we have only to determine, by the method of art. 3, a square matrix |j^j|, and an 
oblong prime matrix satisfying the equation 
the constituents of «! are then the terms of a set of fundamental solutions. 
Or again, if in art. 4 we employ, instead of the inequality (19.), the equation 
- -i 

(23.) 
(r-f l)x(n +4 
a 
is also 
it is easily shown that if j ^ ^ prime matrix, 
a prime matrix ; so that, by folloudng the method of that article, we may obtain 
directly a set of fundamental solutions of any proposed system. Only, it will be 
observed, that in this mode of obtaining such a set, we suppose that we can assign par- 
ticular solutions, not only of systems of the form (16.), but also of equations of the 
form (23.). 
Art. 6 . The importance of fundamental sets of solutions in the theory of linear inde- 
terminate equations is evident from the following proposition : — 
“If jjajl represent a set of fundamental solutions of the system (16.), the complete 
solution of that system is contained in the formula 
■yk=m V 
i—^1 2, 3, 
^k {•) 
(24.) 
in which I 25 • • • Im are absolutely indeterminate integral numbers.” 
For it is evident that every set of numbers included in (24.) satisfies (16.) ; and, con- 
versely, if am-\ i,n+m ally solution of (16.), the determinants of the 
