402 Professor Young's Observations 



rithm, whatever it be, by log a ; then we know that all the 

 values of a'^ will be expressed by 



a- = 1- |l+ {la) X + ^^^ + TtI^^' + ^^'l' 

 and also that all the values of 1^ will be expressed by 

 1- 1 +2fo. V-l + '^^!^^^.^+ (?^Z^V+&c. [1.] 



so that, combining this with the expression within the brackets, 

 we have for a^ the following development, viz. 



«'=l+(Z. + 2^x^^).r + '''' + 't'/~" 



1.2 





V • [2.] 



a development which is perfectly general. 



From [1.] we have the following development of unity, viz. 



l=l+2^7r.v/-l+5^ rT2' + 1.2.3 - + ^^-' 



and this is the development we shall obviously obtain for a", 



provided we put ; 7= for x in [2.1, and consider 



k in [2.], which is an arbitrary integer, to take only the values 

 implied in T^ . Consequently, under these restrictions, we shall 

 have 



2A:7r-\Ari 

 Qla-\-2V It's/ —\ ^^ I 



whatever be the value of a ; and therefore when a= e, the 

 Neperian base 



ei+2A'7rV-i= 1 ... r-7= = loff 1> 



agreeably to the theory of Mr. Graves, whose steps I have 

 imitated. 



Now, in order to obtain the results of Euler, we may pro- 

 ceed from [2.] as follows; and no person, admitting the con- 

 sistency of Euler's results with the common principles, can 

 have any objection to this mode of arriving at them : — 



For brevity we may write [2.] thus : 

 A^ A^ 



which, by putting -j- for x, becomes 



