927 



than once in the course of our developments of the theory of com- 

 plete series. Thus the aggregate arising after the expansion of the 

 symbolic binomial in every term of the series in (84') will converge 

 absolutely in a domain («) satisfying the foregoing condition ; and 

 in this case it may therefore be ordered arbitrarily. We do this in 

 such a way that elements involving symbols ^2,i with the same 

 index / — or symbolic powers B,J with the same exponent i, if 

 exponents have not yet been replaced by indices — are collected 

 into one, and then may find the new arising coefficient in a simple 

 manner. First we remark that, if §, denoted a certain number, the 

 right-hand member of (84') would represent the formal expansion of 

 the magnitude ^i^m (^j) at the point I, = x. Now it is a well-known 

 truth in the theory of functions that the formal development of the 

 function ƒ(?/) at the point y = 0, 







may be obtained from that at the point y = x, 



f^y) = Yi — i! ' 







by expanding e^very binomial expression in the latter series and then 

 collecting in the so obtained aggregate terms involving the sa«ne 

 power y^ of y : the resulting coefficient of y^ is the formal ex- 

 pansion of 



/W(0) 

 k! 



at the point x. We intentionally speak of /(9r?«;:?/ expansion, because 

 it may happen that there is no value of y for which the two series 

 converge, the circles of convergence of f{y) lying wholly outside 

 each other. Or, if there does exist a value as indicated, it may happen 

 that the just-mentioneii power-series for the expression /<^^''(0) : k! 

 diverges because the circle of convergence of the function f[z) for 

 the value z = x does not contain the point z^O. The statement 

 applies in any case since the general form, of the expansions is 

 independent of the particular character of the function in question 

 and functions do occur, viz. the integral transcendental functions, 

 for which the two series converge in the whole plane. 



If now we apply the foregoing considerations in the present case, 

 we infer that the required coefficient of %J^ is the formal expansion 

 in a power-series of w:,|J(0): k! at the point x. But {\n^ formal 



65* 



