PROPERTIES of the CIRCLE. 45 



drawn from Q^ to the fame, both when O k is — 0, and when it 

 is a maximum, this equality mufl exift whatever be the magni- 

 tude of O k between thefe limits. 



And in a fimilar manner is it demonstrated, that the fum of 

 the cubes of perpendiculars drawn from P to the (ides of any 

 other regular figure of an odd number of fides circumfcribing 

 the circle, is equal to the fum of the cubes of perpendiculars 

 drawn from Q^to the fame. For if O m, &c. or fuch parts of 

 perpendiculars drawn from A to the fides of any regular cir- 

 cumfcribing figure of an odd number, n, of fides as lie between 

 O and G, be denoted by A, B, C, &c. ; and O n, &c. or fuch 

 parts of perpendiculars drawn from G to the fame as lie between 



f7 T, 



O and A, be denoted by a y b, c, &c. A-f B + C-f &c, to 



4 



n— 1 v 



terms, — a — b — &c. to terms, = -, if ' n — 1 be a multiple 



of 2 by an even number. Alio- A 2 -{-B 2 -f-C 2 -f- &c. to T—r- terms 



4 *" 



4. a . +.* + ct+ &c . t0 n JZl termS; - tt — 2Xr l . and . Ai+B3i 



• 4 



•f O -f- &c. to f^Z_ I: terms, — 03_£3__ c 3__ & c# to .tzL termr 

 4 4 



= — . But if 0—1 be a multiple of 2 by an odd number, A-f- 



B + C-f &c. to n ~± terms, — # — J— - & c . to -^=£ terms, 



4 4 



= £ A- + B' + C*+.&c. to *±I terms, + a* + ^ + & c , . to 

 —^ terms =■ ^=i X r% and A3 + 'B/+Cs -f- &c. to £±I 



terms 



