402 Demonstrations of Stewards 



bya. Then« = - -^. Hence x-= -^ 



Anda:2ll2=4R2PC2-PC*; or PC^=4{l2PC2-a;2R2. 



In like manner, if y denote the chord of the arch 2PCB, it 

 may be shewn that PB'*=4R2-PB3-3/2R2. And so of all the 

 rest. Therefore, PC* + PB*+ <^c. = 4R2-PC«+4R2 •PB2 + 



^c. - ^R>"+pRM^. = 4R2- PC2"+ PB^T^"^. - R^ 

 ~i'^fr^~$^. But (Lemma 4th.) PC^+PB^ + ^c. = 2nR2 ; 

 and (as was shown above) x^ \y^ -^ ^z. = 2nR2. Therefore 

 PC* + PB*+ &c. =4R2'2nR2-P^2-2/iR2 =8/zR*-2nR^=: 

 6»R^ Q,. E.D. 



Cor. Hence, if any regular figure of a greater number of 

 sides than three, be inscribed in a circle ; and the same con- 

 struction remain ; the sura of the sixth powers of the chords 

 PC, PB, ^c. will be equal to 20nR«. 



For, let the arches PC, PCB, «^c. be multiplied by the num- 

 ber 3 ; and let the chords of the arches 3PC, 3PCB, «^c, be de- 

 noted by d, Gj ^'c. respectively. Then (by Cor. 2d. Lemma 2d.) 



-2 2 



the chord of the arch 3PC = d J^S^Sl^-^^^!^^ Substi- 



3 ^^ 



luting for 2Q,C , and PQ,^, their respective values, 16R^— 4PC2, 



J .02 v u ^ 16R2-PC-4PC5-4R2PC 



and 4R^ ; it becomes, d — = 



4R^ 



12R2-PC-4PC=' TT ,2_144R^-PC2-96R2-PC4+l6PC'' 



. .-■■ ■ — — ■■ -^, JriGncG (X —^ ■ ' ■■ ■ ■ ■■ ■ — — ' .-■■ - ..— — - \ 



4R" 16R4 



and 18a^^R^=144R*-PC2-96R2-PC'»+16PC«; or iR^-PC^- 

 GR^ -PC^+PC^ =cZ2R^ ; whencePC«=6^^R*-9R4-PC2-h 

 6R2-PC^ 



In like manner, it maybe shewn that PB« =e*R4-9R4-PB2 

 4-6R^*PB^. And so of the chords of all the other arches. 



Therefore, PC^+PB^^ &ic. = R^.^+Ti'ZjTfc^- 9R4. 



PC^-f-PB^ + Stc + eR^-PC'^+PB*-!- (^c. But (Cor. Lemma 

 4th.) (Z=»-|-e2+ (^c. =2nR^; andPC^ + PB^^ 4^0. = 2nR2 : 

 and (by the Proposition) PC^+PB^-|- 4'C. = GwR*. Therefore 

 PC« -H PB« + ^c. = 2nR^ — 1 8nR® -f- 36»R*' =20?iR^ 



By the same process of reasoning, when the inscribed figure 

 has more than four sides, it may be shown, that the sum of the 

 eighth powers of the chords PC, PB, <^c. is equal to 70wR*. 



