ME. H. J. S. SMITH ON SYSTEMS OF LINEAE 
21>S 
If 
a 
L-epresent any matrix of the type r X (wi+J"), the determinants of which 
are not all equal to zero, and if >^ 1 , Xj, ... be integers which satisfy the equations 
'' «. . 0,1 
’’ (18.) 
while 2 , a. 
+ 1,3 
J = l, 2, 3 ... rj 
are integers satisfying the inequality 
V* = >« + l- >A 
^/f= t<'/-+i,/f ''•* 
(19.) 
the determinants of the matrix 
(r+l)X(m+r) 
tions (IS.), we may express each of the determinants 0X 
linear function of the determinants of 
in succession as a 
are not all equal to zero. 
For if A- it is e%ident that by combining this equation with the equa- 
rx(w+r) 
a 
j(r+l) X (w+r) 
II « 
minants do not all vanish, neither can the latter. 
Let, then, ... represent any particular solution (other, of course, 
tliaii that in which every indeterminate is equal to zero) of the system (16.) ; and let 
If, therefore, the former deter- 
n+ 1, 29 
... A 
n + 1, n~\-m 
be integral numbers satisfying the inequality 
^lc=n + m A ^ 0 • 
1 + ) , & k ^ > 
the svstem 
A;_ , A’,+A;^2^’2+ ••• 0 
^=l, 2, 3, .. . n-\-l 
( 20 .) 
(21.) 
terininateness is therefore m— 1. Let 
the same principle, that 
represent a complete set of 
■om a second application of 
represents a complete set of independent solutions 
(which is obtamed by the addition of a single equation to the system (16.)) is itself an 
independent system, as appears from the principle just enunciated ; its index of inde- 
(m— 1) X (7i+TO)j 
a 
independent solutions of (21.); it may then be inferred, from a second application of 
»^X(?^^-m) 
a 
of the proposed system (16.). Thus the determination of a complete set of independent 
solutions of a system of which the index of indeterminateness is m, depends on the 
determination of a similar set of solutions for a system of which the index is lower by 
a unit. By successive reductions, therefore, we shall at last arrive at a system of which 
the index of indeterminateness is unity, the complete solution of which is of course 
immediately found by evaluating the determinants of its matrix. 
The practical application of this method supposes only that we can always assign a 
j)articidar solution of a system of the form (16.) or (21.). And this, it may be observed. 
