A PARTICULAR METHOD IN CENTROIDS 223 



r being the radius of the sphere and 2^ the angle of the cap. 



For a hemispherical shell the distance from the center is— (7) 



From this last result (7) the Center of Mass of a lune of in- 

 finitely small angle may be immediately deduced by considering the 

 hemispherical shell to be made up of such lunes. If x be the dis- 

 tance of the Center of Mass of one of the thin lunes from the center 

 of the sphere, the Center of Mass of the hemisphere is reducible to 

 that of the semicircular arc, radius a?, formed by the Centers of 

 Mass of the lunes. Its distance from the center of the sphere is 



2t t 



•'• ~ by (2); but we know this distance to be -^ by (7) 



1x 



or the distance of the Center of Mass of an infinitely thin lune from 



IT T 



the center of the sphere is — ( 8 ) 



4 



From this we get the Center of Mass of a lune of any angle 2^. 



TT/' 



Beiner the Center of Mass of an arc Z 2^ and radius — its distance 



TT/* Sill u 



from the Center of the Sphere is by (1) (9) 



It has been thought desirable to mention these results in pass- 

 ing, as being capable of easy determination without the use of the 

 method at present under consideration. The case of the hemispher- 

 ical shell must apparently be found independently of this method. 



5. It will be interesting to investigate one of the foregoing 

 results e. </., the Center of Mass of a lune of infimtely small angle 

 by the method of the present article or rather by a reversal of it, 

 using as a necessary assumption the Center of Mass of a hemispher- 

 ical shell. 



Let ABC (Fig. 3) be a section of a hemispherical shell through 

 its center, and perpendicular to its plane. Its Center of Mass G is 



