82 Mr Brommch, 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/^jr (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)/yjr (r), where h (r) is 

 an integral function of r, in general of degree one less than that 

 of yjr (r). 



Then let 9 = [y-h (r)]/f (r), 



so that h = y — g^fr ; 



thus 6 (h, r)= 6 (y — gilr, r) 



= e{y,r)-gf d ^+... + (- 1)% (gff 



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, yjr 



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 



6{h{A), A) = R (A)yfr(A) = 0, 



for we have yjr (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 — J.) _1 /(r) for the points r = a, b, c, ...; for as 

 proved above h(A) is the sum of the residues of (rE — J.) -1 /?, (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)- 1 /^) and of (rE- A)- 1 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)=^f(a)A 1 +f'(a)A 2 +...+ f ^^ l A c 



just as before. 



With regard to the priority of discovery it may be sufficient to 



