So INVESTIGATION of fomc 



And generally when m is any integer whatfoever, we have 



Ztf #2 — j =5 ==3 



+ &c. equal to 2 « r* + -.rxO + O/+O/ + &c + 



w OT — i "*— 2 m ~3 v O 4- 6? + 65 + &c. w «*— I ^—2 



— • " • • X. ± ! I . — .. 



I i 3 4 r I i 3 



^ ^_4_ ^ x 5< + 6/ + 6d + Sec. &c _ _ (c . 

 450 r3 . / 



AP + BP 4- CP + &c. AQ^ + BQ^ 4- CQ^ + &c. 



d m r m ~* d m r'"~i » 



which, when the circle is equally divided in the points A, B, C, 

 &c. by the circumfcription or infeription of a regular figure, 

 coincides with the 36th and 38th of Dr Stewart's general 

 theorems. 



And univerfally if m have to / any ratio whatfoever 



m ' m + ^ ~ + OCC. 



7 — 3 1~ 3 7"~3 



r r r' 



is=2»rH- j—j-' x °^ +0/ +°^ +&c. + 7'— 7 ? . 



•J 



4 ; 4 



«— 3 / v Of +Q/+Oi+&c. , « ^ZZ . CT ~ 2l m —3 l m -*t l 

 __x +/ - l • 3/ -77-77 



*?— 5* x O^ + OZ+O** x8cc > + &c. &c 

 6/ z* 3 



This lail theorem, or expreffion, 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 fumraa- 

 tions. It may alfo be extended to the chords AP, BP, &c. 

 and exprelTed in terms of them. And as to the truth of the bi- 

 nomial 



