68 INVESTIGATION of fome 
See Fig. t. and Theorem ®. Since the part of the tangent 
at the point A, that would be intercepted between perpendicu- 
lars drawn to it from P and Q; is equal to 2 Pa, or 2 Qe, the 
part of the tangent at the point B, that would be intercepted be- 
tween perpendiculars drawn to it from P and Q; is = 2 Pe, or 
2Q /; and the part of the tangent at C, that would be inter- 
cepted between perpendiculars drawn to it from P and Q; is 
= 2P4, or 2Qd, we have (when AB, BC, &c.. are equal, or 
when the diameters pafling through A, B, C, &c. make equal 
angles with one another at the centre O) the fum of the fquares 
of thefe parts of the tangents, (calling » the number of the 
I . 
points of contact), =X =. 2 r’; the fum of their fourth pow- 
I 
ers 2x = x 2? 7+; and the fum of the 2 powers of thefe 
1 2 Te3.5e eo. 2M—2T 
parts (7 being any integer lefs than ”) = 2x LG : =“ es 
cB eit te 
X 2m7 (r being the radius OP or OQ) = the fum of the 2m 
powers of the chords drawn from either P or Q, at right angles 
to the diameters paffing through A, B, C, &c. = the fum of the 
2 m powers of chords, drawn to any point in the circumference 
from the angles of a regular infcribed figure of » number of 
fides, or from the points where a regular infcribed figure of x 
number of fides, touches the circle, = the fum of the 2 7 powers 
of perpendiculars, drawn from P or Q to ” number of right 
lines pafling through Q_or P,,and interfeéting each other at equal 
angles. And the fum of the 2™ powers of the halves of thefe parts 
of the tangents, or of the parts intercepted between the points of 
conta&t and perpendiculars drawn from either P or Q to the fides 
of the equal fided figure circumfcribing the circle, or fegment, is 
1-3.5.+++ 2M—I 
= 
= 1,2.3.+0+ Bie 
x r™ = the fum of the 2m powers of 
the 
