122 Mr. E. W. Barnes. 



to the dependence of a. n upon n. When a n is, for n large, to the first 

 approximation a transcendental function of n, the integral function is 

 said to be one of transcendental sequence. 



A function with a finite number of simple sequences of zeros can be 

 built up of a number of non-repeated functions each of simple 

 sequence, 



When the nth zero of a function of simple sequence is repeated a 

 number of times dependent upon n, we call it a repeated integral 

 function. 



The typical zero may require more than a single number to define its 

 position in the series of zeros to which it belongs ; the integral function 

 is then called one of multiple sequence. 



Finally, it is pointed out that the category to which a typical zero 

 belongs may be indefinitely complex, and that, therefore, we can 

 expect to lay down no general law relating to all integral functions 

 which is not a disguised truism. In the memoir we confine ourselves 

 substantially to simple functions with repeated or non-repeated 

 sequences of zeros. 



Part II contains the theory of divergent arid asymptotic series, 

 Poincare's arithmetic definition is first given ; the function J (i) admits 

 the asymptotic expansion 



in certain regions where \z\ is very large, if the sum of the first n terms 

 be s n , and the expression 



tends to zero as s tends to infinity in those regions. 



The difficulty of this theory is pointed out, and it is shown that a 

 theory dependent on analytic functionality is more tractable and no 

 less precise. 



A series 



of finite radius of convergence can, by an extension of a method due 

 to Borel, be interpreted by means of a contour integral for all values 

 of the variable outside the circle of convergence, except those which 

 lie along the straight line joining the singularities to infinity, and 

 proceeding directly away from the origin, provided all the singularities 

 of the function represented by the series lie on such a line. The 

 integral is called the " sum " of the divergent series. The process of 

 summation is not unique, but must always lead to the same result, 

 namely, to the function which is the analytic continuation of the 

 function represented by the given series when convergent. On the 



