obtained by Means of imaginary Quantities. 
103 
3. The sum of cos. x -f- cos. 2 x -f- cos. 3 x . . . . + cos. n x 
co ^-cos ■ (» + Qx+cos.*— ■ for this sum ; s 
2(1— COS. 
iV 1 - 1 21V-1 , — 2 V" — 1 nxS/— 1 . — r\/. 
+ * 
X K''' -1 + « 
2 
2 *\/' 
■f &c 
+ * 
‘ + 
+ ix 
r\/“ 
W - 1 _|_ &c . 
,—nxV~ 
Now, according to the explanation that has been given of equality, 
e xV-i _^ e zxs/-x 
e .W-x __ e x\/- 1 x ( ! _|_ I e zx V- 1 . . e 
if S were an abridged symbol, which, developed according to a 
n — 1 jtV' - 
certain form, became the series e 
then 1 + S — e 
xv/: 
l -+e 
zx 
v ~ 
. e 
nx 
V- 
n x s/ — 1 Y^Quitj truly represent 
1 ... e^ n - x ^ F 1, and therefore S 
and e* -1 x(i-f S — .r v — 1 j b e expressions equally 
significant, or S would = e xS// ~' 1 Se* ^ ~ 1 — e^ n + *) * ^ ~ \ 
or S 
ft + I ) X V — I xV — I 
: , the form that must be given 
e x ~~~ 1 — 1 
to S ; so that, when expanded after a known form, it becomes 
e z — 1 e zxS ^ — 1 e nx ^ — 1 
In like manner, the abridged symbol for e~ x V~ 1 -f- e~~ 2 x v/ ~ 1 ... e~ W -*V'— ! 
but index n— zm-\- z — o.\ zm = n+z.‘. the coefficient Is — — 
(n — 2) n 
1 • 2 • 3 • • • ~ ~ 
_ ( ,1 + 2 1 (” + 47 - (2?! — 2) 2 "n .... _ . . _ 
— „ . (n_ 2 ) (71 — 4) 4.2 5 ant * 11 1S t " e greatest coefficient, since the coeffi- 
cients of the adjacent terms are determined by multiplying it by ^ ^ and 
ft - — m -f- 1 
respectively ; or, since m — 
+ 2 
by -i- and -j-. 
1 n+2. n + 2 
