obtained by Means of imaginary Qiiantities. 101 
i. Sin. x . cos. y — \ sin. ( x + y) + i sin. ( x — y). 
. e x -^ - l - e- xV - 1 A v - 1 + g— - 1 
For sin. x = , cos. y — _ 
2 _ I ^ 2 
■4 
jr v' — i — V" _—i 
— e 
/. sin. .r x cos. y 
v ?y- 
= by what has been shewn, 
,yY - 1 + 
3 
2^ 
x(« 5,v/ - , + - vv/ 
4 _ I 
y v _ i 1 
( y V — i — y V — il 
X [eS . + e J 
2 V — I 
e (* +y) y I . e {x.r-y) - I -(x— y) V' - - 1 e _ (x + ;y) f - i 
\ X — 4 r X 
2V - 1 2^ — 1 
*(■* + :0 _ tf — (x _ g - (X + j) v'lTT 
-x — = z * 7= 
2 V — i 2 V — 1. . 
= \ x.sin. (x -}-y) -{- x sin. (x — y\. 
2. cos. x ==— — (cos .nx -f- n . Cos. [n — 2) x -fc- ,f r . 'J cos. (» — 4) .r -f &c* ) 
A — x y .-— 3 - )- — + ^ cos. (£.)' [n an odd number) or 
= (cos. nx 4- ?z . cos. (n — 2) jr -{- j- 1 - cos. [n — 4) .z -f & c -) 
(” + 2) (« I 4) 2 n T 
n . (n _ 2) . . . . 4. . 2 
~x '4“ 
x 72 being an even number: 
for cos. x — 
T _ .r 4 - 1 n / rV i + 7- x ✓ - 1 V 
_i_ ,.COS.X-^ ±_ J 
a/ 
+ e 
x V — 1 
"- 1 f. 
— r v' - 
; now. 
if ( ^■ r ' A/ — 1 _j_ >- x </ _ ij» - 1 W ere = ^(« — 0 <*■ v — * _j_ [n — i j 
e't n 3) x-'/ — 1 &c. [e x ^ ~~ 1 -j- x v “ ‘j” would == e t ! xV '~ 1 
_j_ ?2 — 2) x _ 1 _|_ & c> or, if the developement of 
[ exS ~ 1 + e ~ xV ~ i ) n ~ 1 were according to the law of the 
binomial theorem, the developement of [e x v ~~~ l + 
would be according to the same law; but the developement of 
