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) = ^ a (s) 
6 (a) + 6' (a) (s - a) + .. + ^ ^ (s - a ) a ~ 1 
(a -1)1 
where s is any quantity. 
But 
fa(s) 
fla-1 ( a \ 
0 (a) + O' (a) (s - a) + . . . + ^ ^ ( s - a) 0 
(a-l)l 
is the residue at r — a of the function 
6 (r) ( s - a) a f(r) f ( s ) 
' (s — r) (r — a) a (s — r)ty (r) ' 
We have accordingly to consider the sum of the residues of 
this function at r = a,h,c, ; now the residue of f(r)/(s — r) is 
zero for each of these points (assuming as before that r = a,b,c,... 
are not singular points for f(r)) and thus we may replace H ( s ) by 
the sum of the residues of 
/(r) f(s)-jr(r) f(r) 
■f(r) s-r f(r) XK ' h 
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 )-> -f (r). 
Now F ( 5 ) is a polynomial in s, and H {A) is therefore found 
by writing A for 5 ; 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 f(r) {rE — A ) _1 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), C (s), ... the 
residues for r = a, b, c, ... of the function 
( r ) - 't 
-f (r)(r-s) ’ 
1 American Journal of Math ., vol. 16 (1893), p. 123 ; Math. Annalen, Bd. 46 
(1895), p. 561. 
2 Grelle’s Journal , Bd. 84 (1878), p. 54 (§ 13); the investigation given above is 
taken from the Berliner Sitzungsberichte , 1896, p. 604. 
