A PARTICULAR METHOD IN CENTROIDS 



227 



The geometrical investigation of limiting values has been 

 avoided, it is hoped successfully, by using a rotation as equivalent to 

 actual transfer of the infinitesimal portion. In the present example, 

 however, it seems advisable to go somewhat into detail. 



D (Fig. 4) being the intersection of AB and OG, AD = DB; 



a, 

 the Z DOM=^ being infinitesimal, DM is infinitesimal, .-. AM 



=MB ; similarly A' M = MB' ; .-. the centers of the circular sections 

 AB and A'B' are at M, .-. all circles on the surfaces of the wedges 

 MAA', MBB', passing between A and A', B and B', and through 

 the vertices of the wedges, have in the limit the common center M : 

 their radii, though not absolutely equal, differ by infinitesimals from 

 r sin 6 .*. in the limit, the volume and Center of Mass of each 

 wedge may be taken as if it were cut by diametral planes including 

 an infinitely small angle, out of a sphere of radius r sin 6. 



8. If a circular lamina of uniform m,ass, be gradually 

 'pushed over another of the same diameter arvd thickness^ to find 

 the Center of Mass of the portion of either uncovered hy the other ^ 

 when they are just ahout to coincide. 



Fig. 5 



