INACCURACY OF SOME LOGARITHMIC FORMULAE. 183 



whereas [32], on supposing R = 1, S = 0, A = e, and B = 0^ gives the more 

 general formula 



-7=— [Note K.l 



A remark necessary to prevent misconception is, that, in certain cases, a 

 logarithm may re-appear at intervals with different ranks in different orders. 



NOTES. 



Note A. — My knowledge of these researches is derived not from the original Essays, but from 

 abstracts of their contents given in the Dublin Philosophical Journal, vol. ii. No. 3. p. 60. and No. 4. 

 p. 219. 



My occupations have prevented me from examining whether mathematicians have directed further 

 attention to the extended application of the principles there promulged. In October 1826 I had 

 obtained the results presented in this paper. 



Note B. — As long as the development [18] is not illusory, its values will be independent of the 

 value assigned at any time to the arbitrary constant c. [Vide infra, Notes E and K.j 



Note C. — It is important to observe, that notwithstanding the infinite number of values of f "" a, 

 yet where a; is a real and rational quantity, ^ or f (x f ~ a) will, from the form of the function, have 

 periodical recurrences of the same values. 



Note D. — When this expression is required to assume particular values, there needs be no corre- 

 spondence between the numerator and the denominator ; for, y being supposed for a moment given, 

 X, by the definition of " logarithm of y," may be any such quantity that y may be found among the 

 values of a' or (see [20] ) f (x f ~ a). Every value whatever of formula [22] satisfies this criterion ; 



for, let ^ _ J be any one of its values, in which the numerator and denominator are wholly inde- 

 t a - _ 1 



f _A 



>— 1 f — 1 



pendent, then will a ° or f ( ^_ j f a) possess among its values 



f a 



f{^-f-'«)=ff-^ = . 



Note E. — As this example seems to lead to the general consideration of diverging and illusory 

 series, I shall endeavour to state succinctly my impressions respecting that important and delicate 

 subject. 



Instances frequently occur to the analyst of developments, in which, upon substituting a particular 

 value for the variable in each, there is no approximation to numerical identity between the several 



