84 Mr Bromwich, Theorems on Matrices and Bilinear Forms. 



To show that this is the same as Frobenius's formula, let us 

 consider the value of 



H(s) = Xf a (s) 



6 (a) + 0' (a) - a) + ... + p-^Q (s - a)- 1 



(a -1)1 



where s is any quantity. 

 But 



fa(s) 



6 a ~ l (a) 

 6(a) + 6'(a)(s-a)+ ... +*- — ^( s - a )«- 



(a-l)I 



is the residue at r = a of the function 



6(r)(s-aY f(r) + (s) 

 Ta W ( s _ r ) ( r _ a y ( 5 _ r ) ^ ( r ) ■ 



We have accordingly to consider the sum of the residues of 

 this function at r = a, b, c, ... ; now the residue of f(r)/(s — r) is 

 zero for each of these points (assuming as before that f = a,b,c, ... 

 are not singular points for/(r)) and thus we may replace H (s) by 

 the sum of the residues of 



f(r)M*)- + (r) _f(r) 

 ^(r) s-r ^(V) 



at r = a, b, c, .... Here % (r, s) is a symmetrical polynomial in r, s 

 and has the property that 



X (r, A) = (rE - A)' 1 yjr (r). 



Now H (s) is a polynomial in s, and H (A) is therefore found 

 by writing A for s ; we have just obtained a value for H (s) which 

 has a meaning when s is replaced by A, so we can write 



H (A) = sum of residues of/(r) (r^ — A)~ x for r = a, b, c, .... 



Hence Buchheim's definition of f(A) is equivalent to Fro- 

 benius's; though it may be remarked that the actual calculation of 

 this formula is certainly much more tedious than that of Fro- 

 benius's. 



In explaining Taber's extension of Sylvester's formula 1 it will 

 be convenient to consider first some forms derived from A and 

 given by Frobenius 2 . Let us denote by A (s), B(s), G (s), ... the 

 residues for r = a, b, c, ... of the function 



ty{r)-ty Q) ^ 

 yjr (r) (r — s) 



1 American Journal of Math., vol. 16 (1893), p. 123 ; Math. Annalen, Bd. 46 

 (1895), p. 561. 



2 Crelle's Journal, Bd. 84 (1878), p. 54 (§ 13); the investigation given above is 

 taken from the Berliner Sitzungsberichte, 1896, p. 604. 



