47 



we here again have a case of x oo, correctly interpretable 

 by the left hand member of the equation ; that is, the right 

 hand member, when x = 0, is accurately 1 . When x ■=. w, the 

 signs of the series all become minus : therefore the true value 

 in that case is — 1. 



Before concluding this subject it may be proper to observe, 



that the investigation, whence the series for ^ is usually de- 

 duced, is deficient in generahty. Whenever logarithms are 

 employed in connexion with imaginary quantities, the imagi- 

 nary forms of the logarithms, as well as the real, ought always 

 to be introduced into the investigation : hence the logarithmic 

 expression, from which the series alluded to is derived, should 

 be written thus : 



log?<=M— w-^ ^ 1 ^ -1 t-&c.+2A7rv/ — 1 



By substituting in this e^^— ' for m, and then dividing the re- 

 sult by 2v/ — 1, we shall have the correct and general form, 



X . sin2.^■ sinS*' sin 4 a; ^ , 



- r: sm « h — ^ h &c. + kir, 



where k is any whole number, positive or negative, deter- 

 minable in any particular case, so as to conform to the first 

 member of the equation : regarding that first member, x not 



exceeding tt, as indifferently either -, or kir + -x. 



I have here used the limited logarithmic forms of Euler, 

 and not the more general ones furnished by Mr. Graves's 

 theory of imaginary logarithms,* since these limited forms 

 are sufficient for all the real values in the general result. 



It now merely remains to be shewn that, as briefly stated 

 at page 43, the differential theorem is inapplicable, not only 

 when the proposed series is divergent, but also when it ceases 

 to be convergent, and becomes what Hutton has called a neu- 



* Philosophical Transactions, Part I. 1829, 



