INACCURACY OP SOME LOGARITHMIC FORMULAE. 173 



Postulates. 



To obviate the necessity of interrupting the course of the argument here- 

 after, it may be satisfactoiy to enumerate the principal truths immediately 

 connected with our subject and not immediately evident, which will be taken 

 for granted in this paper. 



For their support, the authority of Dr. Lardner's Trigonometry, Part III. 

 Sections 1 and 2, may be referred to. 



Euler's development of f^, or . 



ffl = i + ^^fi... + (^^^H^... [7] 



1 .2.. .71 



De Moivre's theorem, or 



f(^9) = avalueof(ffl)* [8] 



f" ffl = 2i»r + 9 [9] 



De Moivre's theorem as extended by M. Poinsot, or 



f{x(2zV + «)} = (f«)' ... , t^^] 



Subsidiary division. 

 1st, To possess a development of f ^. 

 The development of Euler [7] is accurate and sufficient. 

 2nd, It remains to obtain a development of f "' ^. 

 Differentiating [6] we obtain 



-^ = V''^ (cos 6 + i/^sin fl) or V~^ d [ ^ 2] 



Substituting in [12] f "' ^ for 0, we obtain * ^.ftfqof':'?-. 



^^^ " = ^^ff'fl ; or since ff~'^ = ^ ; -iV= ^^^«- 



df"^9 ' df^fl 



Hence we find 



df"*« 



t 'ififbiomn .f jt-Ia 

 It is evident by [13], that when 6 becomes = 0, . becomes infinite, and 



