Mr Bromwich, Theorems on Matrices and Bilinear Forms. 81 



where A 1} ..., A a , B lt ..., Bp, ... are certain bilinear forms which 

 can be calculated without any great difficulty when (rE — A)~ l is 

 written out as the quotient of two determinants. It follows that 



/(4) = S 



f(a)A 1 +f\a)A 2 +...+ f 0^ ] A a 



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 



0(y> r)=p y k +Piy k ~ 1 +-.+Pk=0 



in which p , p 1} ..., p^ 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 yfr (r) = 0) are not such as to make the y-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. 6 



