32 INVESTIGATION of fomc 



third proportionals to r, the perpendiculars from either p or q to 

 the fides of the regular circumfcribing figure, and the cubes of 



thefe perpendiculars is equal to 8 /z r 3 + 24 « r x 0/> + $n x 



— ; or, fpeaking algebraically, eight times the fum of the fourth 

 r 



powers of perpendiculars from either p or q are equal to 8 n r\ 

 together with 24 times a multiple by n of the fourth power of 

 the line whofe fquare is equal to r xO p, together with thrice a 



multiple by n of Op . This is Dr Stewart's 26th theorem. 



In like manner, 



" S _±r— 0_^ , r + o/ + r— o/ _j_ r+Ob ■ 



0?Vo£ 4 +0% 4 -f&c. 



are = 2/zr»+20rX0i +0^ +0h +&C+10X 



■t 



— 2nr* •+• 10 nrxOp 4- — * . when the circle is equally 



divided in the points A, B, C, &c. or when a regular figure cir- 

 cumfcribing it touches it in thefe points. 



And generally when m is any integer whatfoever, 



r+Oi +r— Oi , r + Qg +r — Og r + Q£ +r — Oh 



-^=5 "+" r ««— 3 ~T~ r '«— 3 



f» W — I FT- u-rT . r*A c, , m m ~ 1 



+ &cis = 2«r3 + -- T -^xO. + Q? + 0^ +&C.4--— . 



?n _ 2 ?Ji ,.'. of H-^ + O^-f-'&c. » ,g-i; ^.£=3. 

 — • T r 1 1 3 4 



m — 4 m — 5 . Of +Og +Ob + &c , & c . &c. : which, 

 • — z~ X into " 



when the circle is equally divided in the points A, B, C, &c. or 



when 



