Properties of the Circle, 405 



In like manner, CP"4 + BP'^+ &ic. = nR4+4nR2-P"0»4- 

 ftP 0\ Q,. £. D. 



Cor. Let m be any number less than n ; the sum of the 2mth 

 powers of the lines will be nK''"'-^nx^k^K^"'-^-\-nx*\i'^i\-"'-* 

 + nx^C'^W''"-^-\-SfC. in which x denotes the line drawn from 

 the assumed point to the centre of the circle, A the co-efficient 

 of the second term of a binomial raised to the mth power, B of 

 the third term, C of the fourth, <^c. 



Proposition III. 



LET there be any regular figure, of a greater num- 

 ber of sides than three, circumscribed about a circle ; 

 and from any point in the circumference of the circle, 

 let there be drawn perpendiculars to all the sides of the 

 figure ; twice the sum of the cubes of the perpendicu- 

 lars, will be equal to five times the multiple of the cube 

 of the radius of the circle, by the number of the sides 

 of the figure. 



Thus, if n denote the number of the sides of the fig- 

 ure ; and R, the radius of the circle ; twice the sum of 

 the cubes of the perpendiculars, will be equal to 5nW, 



DEMOJVSTRATIOJV. 



Let O (Plate III. Fig. 7) be any circle, and A, B, C, D, &c. 

 the points of contact of the sides of any regular figure, of more 

 than three sides, circumscribed about it. Let P be any point 

 in the circumference ; and draw PM, PN, ^c. perpendicular to 

 SI, IK, fyc. the sides of the figure. Draw the diameter PQ, 

 and also PC, PB, <^c. to all the points of contact. And from 

 all the points of contact, draw CG, BH, 4'C. perpendicular to 

 the diameter PQ. 



It may be shewn, (as in Lemma 4th.) that PM = PG ; and 

 PN=PH ; and so of all the other perpendiculars. But by the 



nature of the circle, PU=^FG=^-=ty Hence PM^ = 



PQ^ 2K 



?5! In like manner, PN'=:— — . And so of all the other 

 8R3' 8R2 



perpendiculars. Hence, by addition^ PI\P + PN' -h &c. = 

 PC^+PBM.j,c. g^^ ^^^^^ p^^p j^ p^, _^ pg, ^ ^^^ ^ 



8R* 



Y 



