32 INVESTIGATION of fome 
third proportionals to 7, the perpendiculars from either p or g to 
the fides of the regular circumfcribing figure, and the cubes of 
thefe perpendiculars is equal to 8r? + 24"rx Op +32xX 
—4 
Me ; or, {peaking algebraically, eight times the fum of the fourth 
; 
powers of perpendiculars from either f or g are equal to 8 2 74, 
together with 24 times a multiple by z of the fourth power of 
the line whofe {quare is equal to r xO , together with thrice a 
multiple by 2 of Of. This is Dr SrEwart’s 26th theorem. 
In like manner, 
SR ee ree eg heey a eee ae 
rtOi +7—O1 470g +r—Og 4 7T4+Oh +7—Oh 
2 r? r2 
= + &e. 
sa ——z —2 Oi +Or' 402 4.&c. 
are = 2nr?+-20rxKOz +O +O+ peer te TE 
A at 
=2nr*?+102r xOp $78t08 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. 
Anp generally when m is any integer whatfoever, 
ry O71 br Oi rFOE +'P Og ~ eee 4+r—Ob 
r—3 re—3 7-3 
drwy Sa om mat 
ope =e H ets + Oh + &c, += -—. 
$ Bec ig BES rx Or + Og + Pi 
—_—_—e 
—i x54 
m—2 y Oi + 08 + Ob ge cea m— 3 
BE 7 1 I 3 4 
6 st et.) 
Oi +Og +O0h + &e: , &c. &c.; which, 
ey aah aera ANT lind 
paw we aes into 
when the circle is equally divided in the points A, B, C, &c. or 
when 
