704 PHILOSOPHICAL TRANSACTIONS. [ANNO 17q6. 



§ 3. In a lemma also states, that if a be any quantity greater than unity ; then 

 the exponential quantity a"^ is greater or less than unity, according as z is posi- 

 tive or negative : and in either case this quantity approaches so much nearer to 

 unity as z is smaller. So that unity is the limit in magnitude or in smailness of 

 a" according as z is positive or negative. Hence also it is inferred that a* is a 

 function of z of the form 1 + az + bz^ -j- cz^ .... 



^ 4. Let now the exponential a* be proposed to be expressed, in terms of its 

 exponent z. First, 



Let a^ = 1 + AZ + Bz"* + /z' &c. 



Then a'"- = I + 2az + 2'bz- + 2^cz' &c. 



a^~- = 1 + 3az + S^Bz' + 3^cz^ &c. 



a"^ = 1 + 4az + 4-Bz^ + 4'cz' &c. 



&c. &c. 



Then by taking continual differences, it at length appears, 



that «-" = 1 + AZ + ^^z^ + :f-^^z' + YT^^z^ + &c. 



A- , A 



-^ -I Z* — &C 



1 . •>' 1.2.3 ' 1 . 2 . . 4. 



A^ 2 , A-t . „ 



+ i.2'.3^' + T:T!:y^^ + &^- 



Thus, Mr. L'H. remarks, that we obtain by a process purely elementary, 

 founded on a property essential and first of geometric progressions, series which 

 have hitherto been deduced for a superior calculus, or at least from the introduc- 

 tion of infinites. 



It is well known, that a is the base of the logarithmic system ; that a is its 

 modulus ; and, making z = \, the relation of « to a is expressed by the equation 



rt = 1 -|- A -I- -^ + J ,_, ^ Making a = 1, the system is that of the 



natural loga-rithms, whose base is denoted by e. From the last series, we can 

 express a in terms of a, either by reversion of series, or by another method to be 

 hereafter proposed. 



■^ 5. To develope the binomial (l -[- a—)", we get 



1 4- AX + - . --— ^ —J- -I- - . — -— . A3 - + &C. 



' 12 n^ ' 1 2 3 h' ' 





=: 1 + AS + -^ A-s'-f "Y", — - A3S3 + &C. 



Now the more that ?? augments, the more the factors 



12 3 4 



1 — ~, 1 — -, 1 , 1 — -, &c. approach to an equality with unity; 



and therefore the greater 7i is, the more the foregoing series will approach to be 



