30 INVESTIGATION of fome 
AnD generally when m is any integer whatfoever, we have 
ae CW; +P =00  FHOf +r—OF r+ Od" 470d" 
aa ps p53 
m—{ 
mM Seed Hiei: ola '32t 1000 
+ &c. equal to anrs+— 7TXOc +Of +Od + &u + 
—+ == ee 
it eT fee a OF TOROS Gc. Me Eee” 
ives 3 4 r Soa 3 
LAG AG PG 
m—3 m—4, m—sy, Oc +Of + Od + &e. ai 
aia aoa (tb x ee a &c. = (Cor. 3.) 
+BP +CP +&. A iQ 
p+ bP +¢ 4, AQ0 4 BQ 4 6Q + be, 
which, when the circle is equally divided in the points A, B, C, 
&c. by the circumfcription or infcription of a regular figure, 
coincides with the 36th and 38th of Dr Srewart’s general 
theorems. 
Anp ‘univerfally if m have to / any ratio whatfoever, 
- m m m 
7pOc%+47—O0" att Of! +r—Of! 740d! 47r—Od" 
Se te Se bE ee &e, 
TRY Tia Be Ta 
r r r 
m m—l m m—l m-2l 
igcanrt x00 +Of + Od + &e. +5: 
yi =: md ¢ 
wal Os 4 OF 0 es : m m—l m—2l m—31 m—4l 
4l or r EAE at a aka 
es a5 
m—s5! Oc + Of +04 K&L Be, &e. 
6/ r3 ; 
Tuts laft theorem, or expreflion, is more general than any 
of Dr STEwART’s theorems, and will furnifh an endlefs num- 
ber of new and curious infinite feries, with their fumma- 
tions. It may alfo be extended to the chords AP, BP, &c. 
and expreffed in terms of them. And as to the truth of the bi- 
nomial 
