202 On the Development of certain Trigonometrical Functions. 



— f tan 5 x tan 5 a: \ ~\ 



.\ 2x*/~- 1 = 2 < tan* — f — &c.J^ \/ — I 



tan 3 .r tan 5 J7 . 



.•. x = tan x - 1 r , &c, + kit 



as before obtained. 



Many other developments, besides this, are deduced in an 

 incomplete form from neglecting imaginary logarithms. Take 

 the following from among many that might be selected. La- 

 croix, at p. 1 36, vol. i. of his Calculus, develops the expres- 

 sion ( */~\) 'v'"- 1 as follows : 



Substitute \/ ^l for u in the known formula 

 log u = u ~ w- 1 - 1 (w 9 -«- 2 ) + i (us-u- 3 ) -, &c, 

 and it becomes 



lo g V"=l = • ~i-^^-4" ( ~ 1 + 1) + t(-*'- 1 



+ 



— 2 it it 



.-. v^llog V~\ = - \ .'.(i/^l) V " 1 =^" s "- 

 This result is incomplete, for it is known that 



* ' 4*+l 



and this is the result which we should have obtained if the 

 imaginary quantity 2 k it V — 1 had not been improperly 

 omitted ; for we should then have had 



log v/^1 = y s/~\ + 2kit ^/^i 



(4/^l)*/-i = e - 



4**1 

 S 



jr. 



These instances not only verify Euler's theory of imaginary 



