82 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 
fractional powers of ( r — a ), contrary to what is assumed below. 
At each of the points r = a, b, c, ... select any one of the k power- 
series which represent y in the neighbourhood of the point 
considered, and expand the quotient y/yjr (r) in ascending powers 
of (r — a ) near r — a, of (r — b) near r = b, etc. Finally, keep only 
the negative powers in each of the series so obtained and take 
their sum, which can be put in the form h(r)jy\r (r), where li(r) is 
an integral function of r, in general of degree one less than that 
of yfr (r). 
Then let g = [y - h (r)]/f ( r ), 
so that h = y — gyjr ; 
thus 6 (h, r) — 0 (y — gyjr, r) 
= 6(y, r) 
where R is plainly an integral function of y, r, g, yjr and so is not 
infinite at any of the points r = a, b, c, .... Further, since h, 
are both rational functions of r, R must also be rational in r. It 
follows from these properties of R that we may substitute A for r 
in R, and then also in the last equation, which gives 
0(li(A), A) = R(A)^(A) = 0, 
for we have yfr(A) = 0. 
Hence h (A) satisfies an algebraic equation of the same form 
as that which defines f(r)\ and we may write accordingly 
f{A) = h{A), 
and take this equation as the definition of f(A). This method of 
proof is slightly amplified from that given by Frobenius for the 
case A* ( Berliner Sitzungsberichte, 1896, p. 7). 
It is easy to see that h (A) is really equal to the sum of the 
residues of (rE — A) -1 /(r) for the points r = a, b, c, ...; for as 
proved above h(A) is the sum of the residues of (rE — A ) -1 h (r) 
at these points. Now we constructed h (r) so that \h(r)—f(r)\ 
when expanded near r = a, should start with a term in (r — a ) a ; 
it follows that the residues of 
(rE — A)~ x f(r ) and of (rE — A)~ l h (r) 
are equal for the point r = a\ the same holds for r = b, r = c, ... by 
a similar proof. It will follow that 
f(A) = h(A)=l 
f(a)A 1 +f'(a)A,+ 
/“-'(«) 
(a — 1)! 
just as before. 
With regard to the priority of discovery it may be sufficient to 
