XXII. Consideration of various Points of Analysis. By John 
F. W. Herschel, Esq. F. R. S. 
Read May 1814. 
On the Calculus of Generating Functions. 
1 N whatever point of view we consider the theory of gene- 
rating functions, whether as the fertile source of new disco- 
veries, or as a medium for exhibiting, in the most comprehensive 
and uniform point of view, results already known ; we shall 
find fresh cause to admire the profound and original genius 
of its author. To the latter of these objects it is, however, 
more peculiarly adapted, and perhaps, in the present state of 
analytics, this may even be considered as the more precious 
advantage of the two. Such has been the indefatigable activity 
of tliose illustrious men., who have devoted themselves to the 
pursuit of mathematical science, that analysis must be consi- 
dered as already adequate to every purpose to which we can 
reasonably hope to see it applied. The attention of the scien- 
tific observer must now be directed to those elevated stations, 
from which distinct and extended views of its arrangement as 
a whole can be obtained. The calculus of generating func- 
tions affords such a station, and commands a wider and more 
magnificent prospect than any which has yet been opened to 
the view of the speculative philosopher. It becomes interest- 
ing then to extend its application as far as possible in this line, 
and to introduce it on every occasion where there seems any 
