Mr Bromwich , Theorems on Matrices and Bilinear Forms. 81 
where A 1} ..., A a , B 1} Bp, ... are certain bilinear forms which 
can be calculated without any great difficulty when (rE — A)- 1 is 
written out as the quotient of two determinants. It follows that 
f(a)A 1 +f'(a) A 2 +...+ 
/—(«) 
A , 
]■ 
0 - 1 )! 
the summation extending to all the points a, b, c, .... 
This definition is readily extended to such transcendental 
functions of A as are defined by power-series, convergent for the 
values a, b, c, . . . ; for our theorem will hold up to any finite 
number of terms ( n ) of the series ; and the right-hand side of the 
equation has a definite limit as n tends to infinity, provided that 
f(a),f'(a), ..., f(b), f' (b), ... have finite limits. Making these 
assumptions it is natural to define f{A) as given by the limit of 
the right-hand side for n infinite. 
Such transcendental functions have been given by Schur 1 , 
Mettzler 2 , and Taber 3 ; as far as I know, the exponential function 
is the only one that has been of use in any investigations of im- 
portance; this function has been used by Schur in the theory 
of continuous groups and by Taber in certain researches on the 
linear automorphic substitutions (of a bilinear form) which can be 
generated by an infinitesimal substitution belonging to the same 
group. 
It may be useful to point out that with the above definition 
for the function exp A, we have 
exp ( mA ) = (exp A) m , 
if m be an integer. The proof of this follows without any 
difficulty from the expression given below (foot of p. 85). But 
exp ( A + B) is only equal to the product (exp A) (exp B) if A 
and B are commutative {i.e. if AB = BA). 
If f(r) be an algebraic function of r , i.e. the root of an algebraic 
equation whose coefficients are rational functions of r, we can 
readily extend our definition so as to obtain f{A) by a rational 
process. Thus, let y = f(r) be an algebraic function of r, defined 
by the equation 
6{y, r)=p 0 y k + p 1 y k ~ 1 + ... +_p* = 0 
in which p 0 , p lf . .., p k are rational functions of r, which may be 
assumed to be integral functions without loss of generality. We 
assume also that the values r = a, b, c, ... (which are given as 
before by (r) = 0) are not such as to make the ^/-discriminant of 
6 vanish ; for, if the discriminant vanished at say r = a, the 
expansion of y in the neighbourhood of r = a would contain 
1 Math, Annalen, Bd. 38 (1890), p. 271. 
2 American Journal of Math., vol. 14 (1892), p. 326. 
3 Math. Annalen , Bd. 46 (1895), p. 561 ; and several other papers. 
VOL. XI. PT. I. 
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