80 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 
can be determined ; the method is, also, capable of being extended 
to functions, which are not rational and algebraic. 
Suppose f(r) any rational function of r and then consider the 
value of the sum of Cauchy’s residues for the function of r 
fir) yjr (r)_- f(s) 
yjr(r) r — s 
at the points r = a,h,c , We see at once that this sum is 
equal to 
f(s) — ( sum of residues of { ^ )_' v / r _( s ) for 
y l _ jr (r) (r-s) 
\ the points r = s, r = a, h, c, ... 
Now considering all the residues of f(r)/[yjr(r)(r — 8y\ we know 
by a theorem of Cauchy’s that their complete sum is zero ; and in 
addition to the points r = s, r = a, b, c, ... there are the poles of 
f{r) and possibly r — x . Thus the original sum of residues is 
f(s) + yjr(s) ( sum of residues of t for\ 
y T l yfr(r)(r-s) J 
\ the poles of f(r) and r= oo / 
=f(s) + f (s) g (s), 
where g ( s ) has the property of not being infinite for any of the 
values s = a,b,c,..., for we suppose that r = a,b,c,... are not 
poles of f(r) otherwise the function of the given bilinear form 
would have no meaning. It will be recognized that g ( s ) is the 
f(s) 
sum of those partial fractions in , ' which are not infinite for 
^ (s) 
s — a, b, c, ... ; so that g (A) has a meaning. 
Now write s = A in the result last obtained (which we may do, 
for the equation is rational on each side) and then, since yjr (A) = 0 
and since g (A) has a meaning, 
/( A) = sum of residues of (rE — A)~ 1 f(r) for 
the points r — a, b, c, .... 
We can write this in a different form, for we have an equation 
of the type 
(rE-A)~' = 
A x A 2 
r — a (i — aj 
B x | B 2 
r—b (r 
+ 
by 
+ ... 4 
(r - a) a 
It 
(r-by 
