278 THIRD REPORT— 1833. 



of triangular numbers to -^. The series 



1 _ 4 ^. 9 _ 16 + 25 - &c. 

 of square numbers to 0. The series of tabular logarithms 



log 2 — log 3 + log 4 — log 5 + &c., 



would be found to be equal to -0980601 nearly. If we should 

 suppose X negative and greater than 1, the original and the 

 transformed series would become divergent series of the first 

 class. 



The series 



loga = (« - 1) - ^ ^ ^ + ^— o-^ - 4 + &c. 



is divergent when a is greater than 2, and convertible by Euler's 

 formula into the convergent series 



—ar- +li-~^~ + 3"-~^3~ +T"^'»~ + *''" 

 or by the method of Lagrange into the series 



n{^a-\)-^ {^a - If + ^ {^ - If - &c., 



which may be made to possess any required degree of con- 

 vergency. But it is not necessary to produce further examples 

 of such transformations, which embrace a very great part of the 

 most refined artifices which have been employed in analysis. 



One of the most remarkable of these artifices presents itself 

 in a series to which Legendre has given the name of demicon- 

 vergent*. The factorial function F {I + x) is expressed by the 

 continuous expression 



(^) \2^x)'ji, 



where R is a quantity whose Napierian logarithm is expressed 

 by 



A B_ , C 



] .2.x 3 . 4 . a;2 "^ 5 . 6 . :t4 ^^'' 



where A, B, C, &c., are the numbers of Bernoulli. The law of 

 formation of these numbers, as is well known, is extremely 



• Fonetions ElUptiques, torn. ii. chap. ix. p. 425. 



