122 STEWARTS THEOREMS. 



or 8Sp 4 = 8nr*+24w*Z* + 3nZ 4 .................. (6). 



This is the sixth proposition. 



The seventh is, for the m ih power of the perpendiculars, what 

 the first arid fifth are for the third and fourth powers respec- 

 tively. It is this : 



^ _ 2m 1 . 2m 3 ... 1 



m.m 1...1 



,_ 

 r m ...... (7) ...... 



DEM. V* = r m - (! - cos $) m d<f>. 



* 



Let = 



= 2 wl+1 ["si 



^ o 



sm 







2 w - 1 . 2m - 3 . . . 1 2m - 1 ... 1 



7T = 2?T . 



2m. 2m -2... 2 'm.m 1...1' 



1 

 m.m 1 ... 1 



Q. E. D. 



If the assumed point is at I distance from the centre, 



This, it is easily seen, will reduce into the following form, 

 .m I I ,, m...m 33.1 



m _ 474 



r 1+&C ' 

 ...... (8), 



which is the eighth proposition ...... (n> m). 



Lastly, let us consider the 2m th powers of the chords in the 

 case of the inscribed polygon: we have already in the third 

 proposition found the value of their sum when m = 2. 



As before, c 2 = 2r 2 (1 cos </>) ; 



/. 2c 2w = 2V" S (1 - cos </>)"*. 

 That is, as we have already seen, 



m . m 1 . . . 1 



