310 
ME. H. J. S. SMITH ON SYSTEMS OE LINEAE 
wliere «i, ... is a particular solution of (56.), jM/,, jW/j, ... indeterminate numbers, 
and |ja|| a set of fundamental solutions of the system 
1 2 .^ 2 + .... + + 0 ( 
r=l, 2, 3, ... n 
(58.) 
AVhenever, therefore, the proposed system is resoluble, its complete solution involves 
ni indeterminates ; but in order that it should be resoluble, a certain condition must 
be satisfied by its coefficients. This condition is, “ that the greatest dirisors of its 
augmented and unaugmented matrices must be equal We shall call these divisors D 
and Do res[)ectively, representing the matrices themselves by ||A|| and ||Ao|. That the 
condition is necessary may be seen by eliminating in turn every combination of oi — 1 
indeterminates from (56). We thus find that every determinant of ||A|| is divisible by 
Do, ?. e. that D is divisible by Do ; but evidently D divides Do, so that Do=D. To show 
that the condition is sufficient, as well as necessary, consider the system 
0 1 a 1 ~ 1 - A,-_ 2 • • • ■ “b Aj-^ n + m ^n + m 0 
i=l, 2, 3, ... n 
. . . (59.) 
and let 
(?n+l) X(w + ?«T-1) 
represent a set of its fundamental solutions. To say that 
(56.) is resoluble, is the same thing as to say that (59.) admits of solutions in which the 
value of a’o is unity; and (59.) will not, or will admit of such solutions according as 
^ 1 , 0 ? ^ 2, 05 • • • • 0 ciO: 01’ tio not admit of any common divisor beside unity. But, by the 
theorem of art. 7, those determinants of ||Qj| into which the column S,^o, Sg, o • • • enters, 
are equal to the determinants of ||Ao||, taken in a proper order and divided by D. If 
D = Do, the detei’ininants of ||Ao||, divided by D, are relatively prime, and consequently 
those determinants of ||6|| which contain o, oi • - • ^m, o ^I’O also relatively prime ; a con- 
clusion which implies that ^ 2 , o? • • • ^m,o are themselves relatively prime, i. e. that the 
system (56.) is resoluble. 
This criterion is not immediately applicable if the system (56.) be not independent, 
e. if the determinants of its augmented matrix ||A|| be all equal to zero. But it may 
* [This Theorem has already been given by M. Ignaz Heger (Memoirs of the Vienna Academy, vol. xiv. 
second part, p. 111). I regret tliat in the abstract of the present paper, which has been inserted in the 
‘ Proceedings of the Eoyal Society,’ no reference was made to M. Heger’s Memoir, with the contents of 
which I was unacquainted, at the time at which that abstract was prepared. M. Heger’s demonstration 
(adapted to the terminology here employed) is, in the main, as follows. (1) If the unaugmented matrix of 
an indeterminate system be prime, the system is always resoluble. For every determinate system, of which 
the matrix is a urdt-matrix, is resoluble in integral numbers ; and we may suppose the given indeterminate 
system to form part of such a determinate system (see art. 10, supra). (2) The equation ||A|| = ||D|| xj|A7, 
in which l|Dj| is a square matrix, having D for its determinant, and ||A'|| a prime matrix of the same tj'pe as 
A||, is always resoluble (see art. 3). We can therefore replace the given system (56.) by a system of 
which the augmented matrix is ||A'||, and which is resoluble or irresoluble at the same time with the given 
system. But if D(,=D, the unaugmented matrix of this derived system is prime; i. e. if Dy=D, the pro- 
posed system is resoluble. (3) That the condition is necessary as well as sufficient may be proved as in the 
text. — Sept. 1861, H. .J. S. S.] 
