412 Demonstrations^ ^e. 



But, as in the first case, Yp*^Yq*+ &:c. tLnd* (by Prop. IV.) 



8 



and (Prop. III.) V/ + \Y+=Aw(^3; and, (Cor. Prop. IV,) 



Yf-it-yq"-^ S^'C.=^nd'' ', and, (Lemma 3d.) Vp+V^ + ^c. 



-nd', and, (^-R^)+ (c?-R)^+ ^c. =n {d-R)\ 

 Therefore, Pp*+P2*+ <^c. ^^nd'^-lQnd^ (<^-R) + 9ndZ^ 



{d-'Sif-tind {d-Kf-\-n {d-Rf=n'—d*-\Q)d^ + lO^^'R + 



8 



9i^-18f^3R+.9£^^K'-4(/4+ l2(^^R-l2fPR?^ MW+d^-^d^R-^r 



6tZ'R2 -4fZR^ + R" =n. R" + StZ^RH — f^^ 



Hence, 6- P/H-P^'' + ^c.=8 • VI1*+ VS^4- ^c. = BwR^ + 

 ^^nd^iV'+'^nd*. Q. E. D. 



Cor. In general, let there be any regular figure circumscri- 

 bed about a circle ; and let n denote the number of its sides : 

 let m be any number less than n ; and R the radius of the cir- 

 cle ; and from any point, (within the figure, if m be an odd num- 

 ber ; but if even, from any point, either within or without,) let 

 there be drawn perpendiculars, to the sides of the figure ; and 

 likewise a right line to the centre of the circle ; and let v denote 

 the line drawn to the centre ; and let a be the co-efficient of the 

 third term of a binomial, raised to the ?wth power ; 6, the co-efj^" 

 cient of the fifth term ; c, the co-efficient of the seventh term, 

 and so on : the sum of the mth powers of the perpendiculars will 

 be equal to nll"'+nXv^R'^~^+n'Qv^R'''-*->r «C««R"^-«-{- ^c. j 



1 .Q 1 "Q-K 



substituting A for ax |^; B, for hx — ; C, for c , and so on. 



Which expression may be shewn to be true, by reasoning for the 

 5th, 6th, 4^0. powers of the perpendiculars, in the same manner 

 as in the two preceding propositions. 



Note. The odd powers of the perpendiculars are confined to 

 the figure, because, otherwise, in summing up, (as in Case 2dj 

 of this Prop.) part of the odd powers of the lines Vp, Yq, ^c. 

 will be affirmative, and part negative ; that is, part to be added^ 

 and part subtracted : and their sum cannot be found. 



