oe 
New Algebraic Series. 285 
Professor Kramp of Strasburg calls the theory of these 
kind of series, or such as has been assumed: under the form 
fa, the theory of numerical faculties. (facultes numeériques) 
and Vandermond reduces them to powers of the second or- 
der. Kramp has given an ingenious memoir concerning 
them in vol. 3. of Gergonne’s Annals. He has shewn their 
application in the higher analysis, not only in the calculus 
Lines, Exponentials, Logarithms, &c. but also in the in- 
tegration of expressions of the following form St"-*e~'"dt 
taken from t=o to (=w . Sy"-!(1—y’)"dy, taken from 
y=o to y=1 &c. These investigations arising from the 
theory of numerical faculties or of the doctrine of series, 
it, deserves the attention of men 
of science in general, and of Mathematicians in particular. 
and from the extensive application of which they are sus- 
ceptible, the subject is deserving of farther investigation. 
It is well known that the most difficult parts of the higher 
analysis can be deduced from the Doctrine of Series in 
common Algebra. Even the whole of the abstruse calculus 
of derivations as given by Arbogast, can be deduced from 
the single theorem of Taylor so nearly allied to the binom- 
ial theorem of Newton, which as we have seen, can be de- 
duced from the above series. M. J. F. Francais Profes- 
Seur a l’école royal de Vartillerie, &c. has shewn this in the 
most satisfactory manner. 
As the Theory of numerical Faculties, has not, to my 
knowledge, appeared in any English publication, and is ve- 
ry seldom to be found in any of the works of the European 
continental Mathematicians, a ly view of it and of its 
notation, may not be uninteresting- 
n the aiaiiinicd a™'"=a(a+r) (a+2r)- er = Saat 
1)r,) a is called the Base of the faculty, rits Difference, an 
mits Exponent. By reversing the orderof the factors, it 16 
