trajJsaCtions of the sections. 529 



If a; be a logarithm of y of any rank in the «* order in the 

 base a, we shall have a^ = y. 



When individualization is required, the author proposes to 

 denote the logarithm of y in the base a, of the i^"^ rank in the 

 «* order by the symbol «-logf y, since the ordinary symbol 

 (log y) yields no information as to the base, the order, or the 



rank. Thus, e being the Neperian base, e-log^. 1 = g ^* ^_ k/~\ 



Having solved these general problems, the author proceeds 

 to affix limits to some commonly received equations, to explain 

 some of the difficulties and paradoxes incident to the subject, — 

 to account for known facts, and to deduce novel facts relative to 

 the equation a* = y, — to apply his theory to other useful for- 

 mulae connected with exponential functions, and to show how 

 far it accords with ordinary notions in a variety of particular 

 cases ; but the limits of an abstract preclude an enumeration of 

 his results. The following, however, may be noticed : 



Let 



1 



^ "^ VT ^^"^ + *^ + 1 - '^(^•' + *^ + 1? - 4r^' 



then we shall have 

 cos^. ^ (r + -/ — 1 *) 



With reference to this formula, it is observable that Z—L. 



_ P 



+ ■ - is the reciprocal of — , ^ ., , and that 



VV—p p vi—p^ 



when * = 0, j9 is equal to 1 , \/ r^, or either, according as */ r^ 

 exceeds, is less than, or is equal to, 1. 



By showing that the commonly received equation {(fY = a"^^ 

 requires to be thus modified (cfy = I'*' a'*'*', and by determin- 

 ing the corresponding individual values of the modified equa- 

 tion, he points out the defect of the reasoning of M. Clausen, 

 of Altona, (noticed by Mr. Peacock, page 347 of his Report 



for 1833,) which seems to prove that a value of e~ " ^ is 

 equal to 1. He takes occasion to enforce the important distinc- 

 tion between the algebra of formulae that are left more or less 

 indefinite and of individualized values. He remarks, for in- 

 1834. 2 M 



