Mr Bromwich , Theorems on Matrices and Bilinear Forms. 77 
invariant-factors are all linear, Darboux’s rule may be at once 
reduced to the following : — 
Let (X — c) be an a-times repeated factor of | A A' — XE | and 
write 
A m =Lt.r< a -™> 
I C rs> Pr* ] 
| Ps tk) , 0 
’r, s = 1 , 2, 
. * = 1 , 2 , 
this representing a determinant of ( n + m) rows and columns 
(according to a notation suggested by Frobenius 1 and Nanson 2 ) 
where %c rs x r y s = A A' — (c -\-t) E and the p s are arbitraries. Also 
write for the determinant obtained from A m by replacing one 
line , . . . , p n m with x ly ...,x n \ rj m is found similarly with 
Vu y n - 
Finally put 
Z m — ^m/ ( A m _i A m )*, 
= ^/(A,^! A m ) 2 , 
and then we shall have 
ii , -XJ^=S(c-X) (X 1 F 1 + ...+Z a F a ), 
where the summation extends to all roots of | AA/ — XZ | = 0, 
and the products on the right are ordinary, not symbolical. 
We treat {A' A — XE) in the same way; we should note that 
the invariant-factors of \A'A— XE\ and of \AA' — XE\ are the 
same. For we have 
A (A' A - XE) = (A A' -XE) A, 
and thus the invariant-factors must be the same by a known 
theorem. Thus we shall find 
A' A — XE = 2 (c — X) (EiHi +...' + H. H a ), 
where c, a are the same as in the last expression. 
Hence we have two substitutions R, S such that 
R(AA' — XE) R' = Xc r x r y r — XE , 
S' {A' A - XE) S = tc r x r y r - XE, 
or RR' = E = SS', 
and (RA) ( RA )' = tc r x r y r = (AS)' (AS). 
Thus if we take the form 
B = Xc^x r y r 
1 Berliner Sitzungsberichte, 1894, “ Ueber das Tragheitsgesetz, etc.’ 
2 Phil. Mag., vol. 44, 5th Series (1897), p. 396. 
