600 MR THOMAS STEPHENS DAVIES ON 



The conclusion just obtained is Dr STEWART'S 40th proposition. If we put 

 =2 it becomes the 29th; and if also in this case r=(>, we obtain the 26th. 

 Also, finally, if n=\, we obtain the 5th proposition. 



[See also note F.] 



PROPOSITIONS XXII., XXVIII., XXXIX. 



Let there be a regular polygon of m sides circumscribed about a circle, whose 

 radius is (>, and let n be any number less than m : then, if from any point in 

 the circumference of the circle perpendiculars Z A t , /A ....... X A m be drawn 



to the sides ofthejigure, we shall have 



S (Z A m ) =* . -= . O , 



1 . J . o . . . . n 



For, in this case, we have the general form of the th power of the perpen- 

 dicular 



r,.n n f , /2kTT 



ZA* = p 



Expand each term, for all values of k from 1 to m, by Lemma iii., and we 

 find from Lemma iv. that all the terms involving the cosines vanish of them- 

 selves, and the expansion is reduced for each perpendicular to its absolute term. 

 As these are all equal, and m in number, we get at once the general form given 

 by Dr STEWART. 



The proposition just proved is Dr STEWART'S 39th. When n=4, it becomes 

 the 28th; and when w=3, the 22d. 



PROPOSITION III. 



Let perpendiculars ZA J5 Z A 2 .... Z A m be drawn from any point Z to the 

 sides of a regular polygon ofm sides described about a circle whose radium is g : 

 then 'if the distance of Z from the centre of the circle be r, we shall have 



For the general form of Z A* is Qr cos -y ~^i '> an d we shall have, as 

 in former cases, all the cosines mutually cancelling, giving the proposition stated. 



PROPOSITIONS XXII., XXIII. 



From any point Z, (rO), draw perpendiculars Z A 1? Z A 2 , . . . . Z A,,, to a re- 

 gular polygon of m sides circumscribed about a circle whose radius is (> . then 

 we shall have 



