DR MATTHEW STEWART'S GENERAL THEOREMS. 601 



For as before, expanding, and recollecting that 6 V 2 , & m are in arith- 



2 7T 



metical progression, having the common difference , the angular functions 



will vanish of themselves from the expression of the sum of the cubes : hence 



3 o r- 

 we have simply p 3 + v for each perpendicular ; and hence again 



- 



which is equivalent to STEWART'S form. 



This is Prop. 23 ; but putting r=^ we have another proof of Prop. 22, viz. : 



2S (Z A m 3 ~)=5mg 3 . 



CLASS II. Regular Inscribed Polygons. 

 PROPOSITIONS IV., XXVII., XLIV. 



Let there be drawn from any point Z (7- 0) lines to all the angular points of a 

 polygon of m sides inscribed in a circle whose radius is p, namely, Z A 15 Z A 2 , 

 . . . Z A m . then we shall have (n being any integer less than m, and t , t , . . . t n , 

 as before) 



The general form of these values is 



Z A* 2 " ={ ? 2 - 2 ? r cos (d-e^ + r'T . 

 Expand and add ; then, since ^, 2 , . . . . 6 m are in arithmetical progression, 



C\ . 



having the common difference , all the angular functions will vanish from the 



sum. Wherefore only the first line of the expansion in Lemma i. will remain, 

 and that one for each Z A. Wherefore the sum of them is m times that line ; 

 and the Proposition 42 follows at once. 



When n=2 it becomes Proposition 27 : and when also, n= it becomes Pro- 

 position 4. 



Dr STEWART remarks, p. 38, that Prop. 2 is a particular case of the po- 

 rismatic Prop. 9. This will be apparent if we consider that the origin is, in this 

 case, the centroid of the angular points ; and hence that all the cosines involved 

 in the mutual expressions cancel among themselves. We shall, however, return 

 to this subject hereafter. 



VOL. xv. PART rv. 7 z 



