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XVI. — j4n attempt to rectify the inaccuracy of some logarithmic formulae. By 

 John Thomas Graves, of the Inner Temple, Esq. Communicated by John 

 Fkederick William Herschel, Esq. V. P. 



Read December 18, 1828. 



From the recent researches [Note A.] of MM, Poisson and Poinsot on 

 angular section, and their discovery of error in trigonometrical formulae usually 

 considered complete, my attention has been drawn to analogous incorrectness 

 in logai-ithmic series. Accordingly, the end proposed in the present investiga- 

 tion is the exhibition in an amended form of two fundamental developments, 

 as the principles employed in their establishment admit of application in ex- 

 panding by different methods various similar functions, and tend to elucidate 

 other parts of the exponential theory. . 



Let a = y. [1] 



It is proposed to exhibit correct developments ; 

 I. Of ^ in terms of a and x ; 

 II. Of X in terms of a and y ; 

 the corresponding formulae hitherto given being incomplete ; viz.* 



I. ^ = H-.la...+ J|l^... [2] 



II. X, when 3/ is positive, = ^-i^^j^ + lj/ j-g-j 



Some authors, for the case when y is negative, have provided for x the for- 

 mula 



V-l{2i + l)7r + l-y p^l 



la L J 



The notation above used will be adhered to, and requires to be explained. 

 i denotes 0, or any integer positive or negative, and t the ratio of the cir- 

 cumference of a circle to its diameter. 1 a is intended to designate the tabular 



* Lacroix, "Traite du Calcul difierentiel et integral:" Introduction, Art. 25, 27, 28, 81. 



z2 



