14 Proceedings of the Royal Irish Academy. 
"We shall now show that if X be a root of the equation A = 0, the 
determinants An and A22 must have the same algebraical sign. 
This readily appears by considering the system of n linear equa- 
tions — 
{Pu - +i?i2^3 +i?i3^3 + &c. + p^,,x^ = 0, 
Pln^l + P2n^2 + Pzn^z + &C. + {p,„, - X) X„ = 0. 
If X be a root of the equation A = 0, any - 1) of these equations 
enables us to determine the Ji - 1 ratios between the quantities Xi, 
X2, &c. ; and whatever equation we leave out, the results must be 
consistent. 
If we omit the first equation, and from the remaining ones solve 
for X2 in terms of Xi, we get 
^11^2 = ~ ^12^1' 
In like manner, if we omit the second equation, and solve for Xi in 
terms of x^, we obtain 
^22*^1 ~ "~ ^21*^2' 
But from the symmetry of A, it is plain that A12 = A21 ; whence we 
have 
AnA22 = AV (3) 
Hence, if X be a root of the equation A = 0, the detenninants An 
and A22 must have the same algebraical sign. As this is true of any 
two of the set of determinants in each one of which the two suj0S.xes 
are the same, it follows that they must all in this case have the same 
sign ; and, therefore, if theii' sum be zero, each one of them must be 
zero. When each one of the determ in ants A &c., vanishes, equa- 
tion (3) shows that all the other first minors of A must likewise vanish. 
Hence, when X is a double root of the equation A = 0, every first 
minor of A must vanish. 
It is easy to extend this investigation to cases in which three or 
more roots are equal. 
In fact, if X be a triple root of the equation A = 0, it is a double 
root of the equation An = 0, and therefore all the first minors of An 
must vanish, as well as all the first minors of A. 
