JVezu Algebraic Series. 285 



iProfessor Kramp of Strasbiirg calls Ihe theory of these 

 kind of series, or such as has been assumed under the form 

 fa, the theory of numerical faculties, (facultes numeriques) 

 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 

 of Lines, Exponentials, Logarithms, &ic. but also in the in- 

 tegration of expressions of the following form (S?"" ' e-'"<Zf , 

 taken from t = o to l=w . Sy"'~^{\ —y'Ydy, taken from 

 y=^o to y=l he. These investigations arising from the 

 theory of numerical faculties or of the doctrine of series, 

 seem more within the sphere of elementary Algebra than 

 those of Laplace, Lagrange, Legendre, Lacroix, Biot, Pois- 

 son, and others, in similar researches in the higher analy- 

 sis, which are generally beyond the reach of ordinary rea- 

 ders. And as the modern improvements in Astronomy and 

 every department of Physics are principally owing to the 

 improvements in this analysis, whatever is calculated to 

 throw any new light upon it, deserves the attention of men 

 of science in general, and of Mathematicians in particular, 

 M. De Stainville, of the Polytechnic School, has given the 

 series in this communication in vol. 9. of Gergonne's Annals, 

 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 frons 

 the single theorem of Taylor so nearly allied to the binom- 

 ial theorem of JMewton, which as we have seen, can be de- 

 duced from the above series. M. J. F. Francais Profes- 

 seur a I'ecole royal de I'artillerie, &ic. 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 short view of it and of its 

 notation, may not be uninteresting. 



In the expression a™'''=a(a+^) («+2r) {a-\-{m'- 



^yr^) a is called the Base of the faculty, rits Difference, and 

 m its Exponent. By reversing the orderof the factors, it is 



