244 Dynamics. 



or, by developing the whole, 



f7i( < 2a 



Cal. 85. anc [ by integrating, we have 



7i (a 2 RZ Z + f flRz 3 i a 2 z 3 + |R2 4 1 2 4 i 2 S )> 

 which, when 2 = 2 R, becomes 



7i (4 a 2 R 3 + V R 4 | a 2 R 3 +8R 5 8 a R 4 y R 5 ) 

 or 



w(Ja R 3 +|R 4 + f R 5 )- 



To find the value of fx' 2 6' d a/, it is not necessary to begin the 

 calculation again, since from the regular figure of the sphere, it 

 is evident that this value will be similar to the former ; we have 

 only to suppose, therefore, that a, which expresses the distance 

 of the plane PQ from the surface, becomes R ; that is, that 

 this plane passes through the centre, it being supposed at the 

 same time to be perpendicular to its first position, and we shall 

 have 



71 (f R 5 - f R 5 + | R 5 ) Or 71 X T 4 5 RS ' 



The two integrals being added together, make 

 7i (fa 2 R 3 +|R 4 -f f|R 5 ). 



Gcom. Since the bulk of the sphere is^XfR 3 or|7TR 3 , and the 

 distance of its centre of gravity from the plane PQ is a + R > if 

 we divide the above result by the product f TTR S x (a+ R) of 

 these two quantities, we shall have the distance of the centre of 

 oscillation and that of percussion ; thus, 



71 R 3 (a + R) 



