300 
ME. H. J. S. SMITH ON SYSTEMS OF LINEAE 
matrix 
(9}i+1)X(m+w) 
a 
are all zero, while the matrix 
7nx(n-\-m)\ 
a 
is prime ; whence, 
by a principle employed in art. 2, the system 
^k=l ik^kyi 
?' = 1 , 2 , 3 , .... n-\-m 
is satisfied by one, and only one system of integral values of i,, I 25 • • • J or, which 
is the same thing, the numbers ••• cim+\,n+m included in the for- 
mula (24.). 
It may be added, that no fractional values of ^25 • • • Im can give integral values 
to a’l, . .. and that the same values of ^i, x^, x^, ... cannot arise from 
different values of li, % 2 -> • • • im- 
The converse of the proposition just established is also true ; i. e. 
“ If the formula 
(24.) 
represent every solution of an indeterminate system of equations, the matrix ||a|| is a 
prime matrix.” 
For if represent a set of fundamental solutions of the indeterminate system, we 
may express the constituents of ||^|| as linear functions of the constituents of ||«||, by 
means of the equations (24.), so as to obtain an equation of the form 
i!«ii=iigxw, 
denoting a square matrix ; whence it immediately appears that ||a|| is a prime matrix, 
and ||[ a unit-matrix. 
Thus, if we apply Eulee’s method for the resolution of indeterminate equations to 
the system (16.), we obtain, as the final result of the process, a system of equations of 
tlie form (24.); and as it is demonstrable, from the nature of the method itself, that 
these final equations contain the complete solution of the proposed system, their matrix 
is a prime matrix. 
If |}«jl and , Jji be any two sets of fundamental solutions cf the same system, we shall 
have the equation 
ll^>ll=i!^!lxiN', 
denoting a unit-matrix. The matrices, therefore, of all sets of fundamental solutions 
are dediicible, by premultiplication with unit-matrices, from the matrix of any given set 
of such solutions. 
Art. 7. If ||ff[j and jj^|j represent two complete sets of independent solutions of the 
same system, the determinants of j|«|| and ||5|| are evidently connected by the relation 
/3 X|(zl= + ^'X|^|, « and /3 denoting the greatest divisors of ||a|| and ||J|j respectively. A 
similar relation subsists between the matrix of the system and the matrix of any com- 
plete set of independent solutions of it. 
Let jjAlj and |[a|! represent those matrices respectively, K and k their greatest divisors ; 
