68 INVESTIGATION of fomc 



See Fig. t. and Theorem O. 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 P a, or 2 Qj:, 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 P e, or 

 2 Q^J J and the part of the tangent at C, that would be inter- 

 cepted between perpendiculars drawn to it from P and Q^ is 

 zr 2 P b, or 2 Qjs?, we have (when AB, BC, &c. are equal, or 

 when the diameters palling 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 n the number of the 



points of contact), r:«X-. 2r l ; the fum of their fourth pow- 

 ers — n x — - X 2* r* ; and the fum of the 2 m powers of thefe 



1.2 



parts \ni being any integer lels than njzznx — i2 -^ 



I • 2- ^« • • • 7/1 



X 2 m r 7 '" (r being the radius OP or OQ) n the fum of the 2m 

 powers of the chords drawn from either P or Q^ at right angles 

 to the diameters pafling through A, B, C, &c. rr 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 n number of 

 fides, or from the points where a regular infcribed figure of n 

 number of fides, touches the circle, — the fum of the 2 m powers 

 of perpendiculars, drawn from P or Q^to n number of right 

 lines pafling through Q^or P, and interfering each other at equal 

 angles. And the fum of the 2 m powers of the halves of thefe parts 

 of the tangents, or of the parts intercepted between the points of 

 contact and perpendiculars drawn from either P or Qj:o the fides 

 of the equal fided figure circumfcribing the circle, or fegment, is 



— n x r n — X r xm = the fum of the 2 m powers of 



1.2.3. • •• m 2 



the 



