CIRCULAR ARC 127 



This relation shows that 



(the distance of c. g. of arc 2 a from center) 



= cos x (the distance of e.g. of arc a from center). 

 Similarly 



(the distance of e.g. of arc a from center) 



= cos x (the distance of c. g. of arc from center), 

 and so on. Continuing in this way, and substituting, we obtain 

 (the distance of c. g. of arc 2 a from center) 



= cos cos cos cos ^ - 

 2 4 8 2 n + 1 



x (the distance of c. g. of arc -^ from center). 



If we make n very great, the value of becomes zero. Thus the dis- 

 tance of the c. g. of an arc 7^ from the center becomes equal to a, the radius 

 of the circle. Making n infinite, we have 



(the distance of c. g. of arc 2 a from center) 



= a cos ^ cos j cos ^ ... to infinity. 



a sin or 

 Now cos = - , 



2sinf 



... 

 so that 



Making n infinite, the value of sin -^ becomes identical with -2- , so 

 that 2 n sin ^ becomes identical with or, and we have 



a a a ^ , sin a 



cos cos cos to infinity = - > 

 248 a 



so that the distance of the c. g. of the arc 2 a from the center is found to 



, sin a 



be a - > as before. 



