DR MATTHEW STEWART'S GENERAL THEOREMS. 599 



SECTION III. INDETERMINATE THEOREMS. 



These are divisible, with a few miscellaneous exceptions already proved by 

 Dr STEWART in the work, into three Classes ; and which we shall take in order. 



1. Perpendiculars on the sides of a regular circumscribed polygon. 



2. Lines drawn to the angular points of a regular inscribed polygon. 



3. Perpendiculars upon lines which pass through a point and make equal 

 angles with each other. 



CLASS I. Regular Circumscribed Polygons. 

 PROPOSITIONS V., XXIX., XL. 



Let there be a regular polygon circumscribed about a circle whose radius is p, and 

 let n be any number less than m ; also, from any point Z, whose distance from 

 the centre of the circle is r, let perpendiculars Z A 1? Z A 2 , . . . . Z A m be 

 drawn to the sides of the polygon: then, if t , t lt t?, . . . t n denote the co-effi- 

 cients of a binomial of the n th degree, we shall have 



Without sacrificing generality, we may somewhat simplify our expressions 

 by taking the line from Z to the centre of the circle as angular origin, and the 

 centre itself as polar origin. Then, for the formation of the equations of the 

 sides we shall obviously have, 



2V fi n 4T A ,, 2(m 



-- U, , 3 = -- t>, , . . . . O m = 



mm m 



or the angles t . 2 ..... 6 m in arithmetical progression whose common differ- 



2TT 



ence is 



m 



Whence, since the general form of Z A* 2 " is 



ZA, = p-rcos( -< 

 l r \ m 



we shall find upon forming the sum of them by the expansion in Lemma ii., that 

 all the terms involving the cosines disappear by virtue of Lemma iv. This sum 

 is, therefore, reduced to the first line of the expansion for each of the perpen- 

 diculars ; and these are all equal, and as many in number as there are perpen- 

 diculars ; that is, the sum is m times the first line of the development, as stated 

 in the proposition. 



