when a regular figure circumfcribing it touches it in faid points, 
gives Dr StEwart’s 37th theorem, fince the fame reafoning and 
mode of demonftration holds good in regard to half the amount 
of this expreflion, whether the points p and ¢ be in PQ, or in 
PQ produced. 
Anp univerfally if m have to / any ratio whatfoever, 
—_—_—_" m m te . m 
740i 4r—Oi? yt tOg Tt + 1-08? tO T+ r—ObT &e 
a ee ee m m . 
a o—3 3 
xr r r 
ly, Bl See m m—J m—rl 
=a2nrs4+7+— -7TXOi + Og +0) 4+&e. + 5-H 
“ Boy oe eee 
¥ 
m—=3l x Of + Og + OF! + &e. mm m—l m—2l m—3l m—4l 
4l r bp RES gs ee aye ee 
nae eS ule Be 
Gl +O Ob. Og Oot Been CBee Bee! 
6/ fe 
_ Tuts laft theorem or expreflion is more general than any of 
Dr Stewarvt’s theorems, and from it may eafily be derived an 
endlefs number of new and curious infinite feries, with their 
fummations. 
Ir is almoft needlefs For me to obferve, that befides thefe theo- 
rems of Dr MatTTHEew STEWART, an unlimited number of other 
theorems, refpecting figures both regular and irregular, circum- 
fcribing and infcribed in the circle, may eafily be derived from 
the foregoing inveftigation, as well as a great number of geo- 
metrical infinite feries, with their fummations. And as to theo- 
rems refpecting given points, right lines and figures either re- 
gular or irregular, given by pofition, and right lines interfeCting 
each other either in one point or in different points in angles 
either equal or unequal, that are deducible from it, they are in- 
numerable. i 
Vox. VI.—P. I. E ‘ Now, 
