VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS, 703 



article 6 ; and by them may the hyperbolic logarithm of 10 be very easily and 

 expeditiously computed. 



f^III. Elevienfary Manner of obtaining the Series by tvhich are expressed Exponen- 

 tial Quantities and the Trigonometric Functions of Circular Arcs. By Air. 

 Simon L' Huillier, F.R.S. From the French, p. 142. 



The use of logarithms, says this author, and that of trigonometric functions 

 of circular arcs, such as sines, tangents, &c. are so frequent in the more ele- 

 mentary parts of mathematics, that these quantities may be regarded as appertain- 

 ing to the elements ; and that their calculus ought to enter into an elementary 

 treatise. 



In Mr. L'H.'s opinion such calculations have not been derived and treated, by 

 former mathematicians, in a manner sufficiently elementary and simple. 



Some mathematicians, he says it is true, and in particular Euler, in his In- 

 troduction, has explained the process in a manner seemingly approaching to ele 

 mentary, but on close inspection too obscure, and founded on the principles of 

 infinites. Mr. L'H. however professes to treat the subject more simply, and in- 

 dependent of such means of imaginary quantities. The method he follows is 

 that of a M. Pfleiderer, professor at Tubingen, and demonstrator of Taylor's 

 theorem, in his dissertation entitled Theorematis Tayloriani Demonstratio. 



In the 1st section Mr. L'H. states as a lemma, the properties of the differences 

 of the powers of numbers, and finally concludes that in general, the first differ- 

 ences of the powers of the natural numbers whose exponent is m, is, 



. m . mm— \_,,inm — lm~1 

 „-"_(«— l)'" = - ?r-' — - . —^ W-^-f- - . — ^— . — ^ n"'-3 &c. 



The mth differences of these powers, which are the m — 1 differences of the 

 first differences, affected with constant co-efficients. So that, if it has been 

 proved that the m — 1 differences of the m — 1 powers of the natural numbers, 



are the constant quantity \ .1 .Z m — 1; and that the differences of 



the same order of inferior powers vanish ; we may infer also that the m\h differ- 

 ences of the mth powers, are equal to m times the product 1.2.3 m — l ; 



or are the product J . 2 . 3 m — 1 ; and that the superior order of differ- 

 ences vanish. For contraction, Mr. L'H. puts A^ (a" ....—«") to denote the 

 p order of differences of the natural numbers, from a to n. 



& 2. In a lemma, states, that the differences of all the orders of the terms of a 

 geometrical progression, form also a geometric progression, having the same 

 ratio as the first series, and having its terms the products of the terms of the 

 first progression by the difference of the two first terms raised to a power whose 

 exponent is equal to the order of that difference. 



So, of the series 1, a, a% a% a^ a" a"-', the mth differences are 



(a — 1)'" X (1, a, a% a^ a* ) 



