Centres of Percussion and Oscillation. 245 



a -f- R 



a* -f 2aR -f R* + f 

 a -{- R 



* 



Hence the centre of oscillation and that of percussion are below 

 the centre of the sphere; and the centre of the sphere cannot be 

 taken for the centre of oscillation or that of percussion, except 

 when its radius is very small compared with the distance of the 

 centre G from the point of suspension. 



If the sphere is suspended by a rod or lamina, and we 

 would have regard to its mass, it will be recollected, that 



we have found - - H -- for the sum of the products of the 



o 1 2, 



particles of such a body into the squares of their distances res- 

 pectively from the fixed point or axis. Now h is what we have 

 represented by a ; moreover, since 



6 = h'f, and & - hf = /, 

 we shall have by substitution, 



3 h'f A' 3 af . 

 3 12 ' 



this quantity and that for the sphere must be multiplied respec- 

 tively by the specific gravities 5, S', of these bodies, if their spe- 

 cific gravities be different ; then by adding the two products, we 

 shall have, 



s j7 + s v/ + s , x (f a , R , + . -R4 + f} RS) 



for the sum of the products of the particles of the whole system 

 into the squares of their distances respectively from the axis. 

 This sum divided by the product of the masses, S a h' f+ S' f rc R 3 

 into the distance of the centre of gravity from the axis, gives the 

 distance of the centre of oscillation. 



