/ 



2B4 }<tew Algehraic Series*, 



The series of the first member, as is well known, is in^ 

 commensurable (which is easily proved, amoiig other meth- 

 ods, by the theory of continued fractions) and comprised 

 between 2 and 3. It is the base of Napier's System of Log- 

 arithms. In representing it by e, as u>ual, we shall have 

 kx kx A^x^ A^x"^ 



' A . . 



If we make e =0^ in which case A will be the logarithm 



of a, according to Napier's System, (or la,) we shall have 

 xla x^l^a x^l^a 

 a^ = l+— 4-^72+7:2:3 + ^"- 



A formula which gives the developement of exponen- 

 tials into series or of any number a, in a function of its lo- 

 garithm. If in this latter formula x be changed into m, and 

 a, into 1 +a;, it will become 



(■+^r='+^i-^+— 172 + 1.2.3 +^- 



But it has been already shewn that 



a; x^ 



(l-l-a;)"'^! +'^7+^(^—1)1 — 2+^c. whence 



ml^i\-\-x) 'm^l^{\-\-x) X ^ x^ 



+ (m-l)(m-2)7yy;^+ 



&c. 



By making m=o, in this last equation, we have 



X X' 



z(i+x)^Y~2"+y-T+*^^- 



A formula which gives the hyperbolic or naperian loga- 

 rithm of 1+a;. in a function of the number x. 



A variety of other important applications of the general 

 principle, will easily suggest themselves, particularly in the 

 developement of circular functions into series, &sc. A 

 branch of analysis of the greatest importance in modern As- 

 tronomy and Physics. 



