TRANSACTIONS OF THE SECTIONS. 523 



tending to illustrate and confirm them. For example, Mr. 

 Graves had found, for the logarithm of unity to the Napierian 

 base, the expression 



which is more general than the usual expression. This result 

 of Mr. Graves appeared erroneous to the author of the ex- 

 cellent Report on Algebra, which was lately printed for the 

 Association ; but it is confirmed by Mr. Hamilton's theory, 

 which conducts to it under the form of a relation between real 

 pairs, namely. 



Log. (1,0) =(04*^: 



and the connexion of this result with that Report was thought 

 to justify a greater fulness in the present communication* than 

 would have been proper otherwise on a question so abstract 

 and mathematical. 



On the Theory of Exponential Functions. By John Thomas 

 Graves, of the Inner Temple, Esquire, A.M. 



In October, 1826, the author of the Memoir, of which the 

 following pages contain an abstract, was engaged in researches, 

 and obtained results, which were communicated to the Royal 

 Society of London in the year 1828, and published in the 

 Philosophical Transactions for 1829, under the title "An At- 

 tempt to Rectify the Inaccuracy of some Logarithmic For- 

 mulae." 



Certain theorems of Newton, Euler, and Moivre were known 

 to establish a remarkable connexion between exponential and 

 trigonometrical functions ; and the corrections made by M. Pois- 

 son and M. Poinsot in formulae of the latter class, induced the 

 author to apply similar corrections to those of the former, more 

 generally than appeared to have been previously accomplished. 

 Accordingly, his original paper exhibited formulae involving 

 arbitrary integers, by means of which he considered that a 

 solution was afforded for various difficulties that had formerly 



• Since this communication was prepared, Professor Hamilton has learned 

 that Professor Ohm of Berlin has been conducted by a different method to re- 

 sults respecting Imaginary Logarithms, which agree with those of Mr. Graves : 

 as do also the results obtained in other ways, by Mons. Vincent and by Mr. 

 Warren. The partial differential equations (b.) have been noticed and em- 

 ployed, for a different purpose, by Mr. Murphy of Cambridge. 



