178 MR. GRAVES ON A RECTIFICATION OF THE 



Hence, bv ^ 1, constants may be dispensed with in the developments ac- 

 cording to formula [18] of f~' yTTC' ^"^ ^ i + K*" 

 We have, therefore, 



and 



Hence r'^'orf-'^^^-t-'j^. 



Hence the Neperian logarithms of K' are 



, — ri-K« , 1 /I-K^^" + \ 



2 J T — -v/ — 1 



2n+ 1. 



[Note H.] [26] 



Corollary. When i and i are both = 0, this expression reduces itself to 



-2-riiiiL\..+_i_(L:iKV:.t'i 



Ll + K^ ^ 2n+ i^l + K'l J 



which is the tabular Neperian logarithm (see [5] ) of K^. Let it be designated 

 by 1 K*. On comparing [25] and [26], it appears, by [9], that — ^— U K* is 

 one of the values of f ~' K^. 



Hence we have the equation 



f(_^3Iu2) = ^2 [27] 



§ 5. To separate the real and imaginary parts of f "' ^. 



6 in its most general form = R + ^— 1 S. — {See Lacroix " Traite" Sgc. 

 Introd. 87.) 



On inspecting a circle whose radius is supposed to be = 1, it will be 



/ obvious that for all arcs whose magnitude lies between t and — -r, the arc and 



sine at any time are either both positive or both negative. Suppose thei*efore 



such an arc to have for cosine the quantity - yr.- — — , then will its sine or 



/ R? S 



±\/ ^- Y^i , jgg = ~7tP=^3 ^s long as the arc and S have the same sign. 



