of the Circle and Hyperbola, 91 



And at G, 

 ^V_l=log(-*/-l) or .-/'-^-•-l. . (f.) 



At A, 2 * </- 1 = log 1 or e 2 * 7 - 1 = 1. . . . (g.) 



From A we may suppose P 2 to run over the circle a second 

 time and then a third time, and so on to an mlh time; so 

 that in the formulas (c), (d.), &c. we may increase the coef- 

 ficient of s/ — 1 in the exponent of e, by 2 m tr without chan- 

 ging the result ; also we may suppose P 2 on arriving at A after 

 the »2th revolution, to proceed along the real branch of the 

 curve so that the logarithm of any quantity X may be repre- 

 sented by tabular log X + 2 mftv — 1. 



6. Perhaps it may not be unimportant to remark, that in 

 describing the properties of the hyperbola, the proper way of 

 enunciating the connexion of hyperbolic spaces with loga- 

 rithms is the one I have used, namely 2 (A C P) = log (C T 

 4- T P) ; for this expression holds good as well for the ima- 

 ginary as for the real branch of the hyperbola, as I have shown 



... . . I APR . CR. 



above, while the common expression p t A t = log ^r is 



quite unintelligible when applied to the imaginary circle, 

 being derived from the consideration of the asymptotes, which 

 are no part of the curve, though usually drawn with it. 



7. There are several other points arising out of the con- 

 nexion of the circle and hyperbola which it would exceed 

 my limits to touch upon here, but which I shall probably 

 make the subject of another article, should it turn out that the 

 preceding results are new. 



8. In conclusion, I may remark that this investigation is as 

 yet incomplete, or imperfect, for the most general formula 

 arising from it is log X = Y + 2 m n V — 1 (Y being the 

 arithmetical logarithm of X), while the most general formula 

 obtained by other means is 



. Y Y + WTlV-l 



lo g x = i+2w-r 



See De Morgan's Calculus, p. 384. 



Port of Spain, Trinidad, 

 April 2, 1846. 



