178 
MR. GRAVES ON A RECTIFICATION OF THE 
Hence, by § 1, constants may be dispensed with in the developments ac- 
2 K 2 i 2 
cording to formula [ 18 ] of f -1 1 — g g and f -1 1 
We have, therefore, 
2K 3 _ 0 ._, /— / l-K 2 , l l ,i_KS 2n +h 
f l + K 2-227r+A 1 Ll + K 2 ’” + 2« (l + K 2 ) + 2?i + 1 (i + K 2 ) J 
and 
1 + K a 
Hence 
, — r/l-K 2 l / 1 — K 2 \ w i .i_K 9 x 2n+1 \ 
~ ^ 1 Ll + K a * '* 2?Al + K*' + 2»+l(l + K 2 ) J 
,_1 TTJ f— 1 2 K“ f— i 
’ 1 K s or f 1 ^ — f 1 
1 + K 3 1 + K 3 
or .* > — : r i — k 3 , i / i_K*x2»+n 
= 2( 2 - ! ), + 2V-l(^... + ^( rTr J ...| 
Hence the Neperian logarithms of K 2 are 
, — ri-K 3 i /i-K a x 2w + 1 -\ 
2^ + 2x/-l| +jrn( I +K 8 ) “'J 
2 i it — a/ 
[Note I-L] [ 26 ] 
Corollary. When i and i are both = 0, this expression reduces itself to 
_ 2 /LdL* . . . . ‘ t i-kh 2,,+1 
“11 + K 3 2rc+iM + K 2 ' 
which is the tabular Neperian logarithm (see [ 5 ] ) of K 2 . Let it be designated 
by 1 K 2 . On comparing [ 25 ] and [ 26 ], it appears, by [ 9 ], that — sj — 1 1 K 2 is 
one of the values of f “ 1 K 2 . 
Hence we have the equation 
f(- V' 1 1 1 ?) = F [27] 
§ 5 . To separate the real and imaginary parts of f -I 6 . 
0 in its most general form = R + »J— 1 S. — (See Lacroix “ Traits” fyc. 
Introd. 8/.) 
On inspecting a circle whose radius is supposed to be = 1, it will be 
obvious that for all arcs whose magnitude lies between x and — x, the arc and 
sine at any time are either both positive or both negative. Suppose therefore 
such an arc to have for cosine the quantity - 7^1 , then will its sine or 
V X it + o 
+ 
R 3 
R 3 + S 3 ~ 
S 
VR 2 + S 3 ’ 
as long as the arc and S have the same sign. 
