228 UNIVERSITY OF COLORADO STUDIES 



Let ADBD' and A'DB'D' (Fig. 5) be two equal circles. 

 The question is plainly the same as to find the Center of Mass of 

 the area D AD'A' (or DB'D'B), when CC (=AA'=BB') the 

 distance between the centers of the circles is infinitesimal. 



Let g be the Center of Mass of DAD' A' and the corresponding 

 pointy' that of DB'D'B. 



The distance PQ intercepted between the two circles on any 

 parallel to AB is evidently constant and =CC', because in a motion 

 of pure translation all points of a body move through equal dis- 

 tances, .'. the area of any thin strip PQQ'P' intercepted between 

 two such parallels =CC' XRR' .'• in the limit the area of DAD 'A' 

 =CC'X the limit of DD',=CC'x2r. 



If now DAD' A' be transferred to the position DB'D'B, carry- 

 ing its Center of Mass from g io g' , the Center of Mass of the whole 

 circle DAD'B moves from C to C as the two circles now plainly 

 coincide, .-. in the limit, 



gq' TT/'i irr 



'99 



CC 2/'.CC' "'' 2 

 But in the limit gg' is twice the distance of the required Center of 



ITT 



Mass from the center of the circle, .-. this distance is — . (15) 



Note the correspondence between the results in this case and the 

 case of a lune of infinitely small angle. 



If the circles overlap in any position we can similarly find 

 the Center of Mass of the tuicovered portion of either. 



If the angle CDC be 26 and O the middle point of CC, the 

 area of DAD 'A' is r" (26>+8in 26') 



20^ ir7^ 



2/' sin (9 r^(2^+8in 2(9) 



... 0,= ^^^^^^. (16) 



• 26' + sin 26 \ J 



On proceeding to the limit (^=0), this gives the same result as 

 above (15). 



