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XVIII. On the Formulae investigated by Dr. Beinklet for the general Term in the Deve- 
laypmefnt of Lageange’s Expression for the Summation of Series and for successive 
Integrations. By Sir J. F. W. Heeschel, Bart., F.E.S. &c. 
Received April 26, — Read June 14, 1860. 
r 
In the Transactions of the Eoyal Society for 1807, Dr. Beinkley has investigated the 
general value of the coefficient of any term in the development of the function , 
and his result is remarkable for the mode of its expression in terms of the successive 
differences of the powers of zero, or of the numbers comprised in the general expression 
A^O". Since that time, in my paper published in the Transactions of the Society for 
1815, “On the Development of Exponential Functions,” I have exhibited other, and 
much more simple as well as more easily calculable expressions for the same coefficient, 
by means of the same useful and valuable differences, and in that and other subsequent 
memoirs, have extended their application to a variety of interesting inquiries in the 
theory of differences and series. It is singular, however, that up to the present time it 
has never been shown that the formulae of Dr. Beinklet, and my own, though affording 
in all cases coincident numerical results, are analytically reconcileable with each other ; 
nor indeed is it at all easy to see either from the course of his investigation, which 
turns upon an intricate application of the combinatory analysis, or from the nature of 
the formula itself, how it is possible to pass from the one form of expression to the other 
so as to show their identity. This is what I now propose. 
Eeferring to my “Collection of Examples in the Calculus of Finite Differences*,” 
will be found the following relation, which enables us to pass from the differences of 
any one power of zero, as 0*, to those of any other, as viz. — 
{ log ( 1 + A) }"./■( AjO"' = 1 ).... (^— w + 1 )./( A)0*"”, 
or changing x into x-\-n, 
{log (l + A)}»./(A)0^'-^”=(.2;+lX^+2)....(a;+?i)./(A)0^ 
As this equation is general so long as negative indices of A do not occur, we may 
change /( A) into , and it becomes 
/(A)O-— (^+l)....(^+n). 0^ 
provided always that contains no negative powers of A. In this for /'(A) sub- 
* Camb. 1820, p. 70. Ex. 6. 
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