Centres of Percussion and Oscillation. 241 



Let AB be the axis of rotation, and through AB suppose two 

 planes PQ, AR, to pass perpendicularly to each other ; let m be p . 

 any particle of the body in question, and having let fall the per- 

 pendicular m F upon AB, we draw m H perpendicularly to the 

 plane RA ; and joining FH, this line will be perpendicular to 

 AB, and consequently to the plane PQ. The right-angled trian- 

 glem IW gives 



- Hm; 

 whence, 



J m X Fm orfm r 2 X 



The problem, therefore, reduces itself to finding the sum of the 

 products of the particles into the squares of their distances from 

 two planes, which pass through the axis of rotation, and are per- 

 pendicular to each other. Now, when the algebraic expression 

 for this sum is found with respect to one of the planes, it is easily 

 obtained with respect to the other. Let us therefore inquire how 

 we can find the sum of the products of the particles of a body 

 into the squares of their distances respectively from a known 

 plane. 



We will suppose the body divided into infinitely thin strata, 

 parallel to the given plane ; and, representing the thickness of 

 one of these strata by DD, its surface by tf, and its distance FDp. r 

 from the plane in question by x, since the points of the surface 6 

 are all distant from the plane PQ by the same quantity x, we 

 shall have x 2 a d x for the sum of the products of all the points 

 of this surface into the squares of their distances respectively 

 from this plane, and consequently f*x 2 <> dx for the entire sum 

 of these products for the whole body. 



If we represent, in like manner, by x' the corresponding dis- 

 tances from the plane perpendicular to PQ, and passing through 

 the axis of rotation AB (the body being supposed to be divided 

 into strata parallel to this second plane), and by 6' the surface 

 of one of these strata, we shall havens/ 2 6' d x' for the sum of 

 the products of the particles into the squares of their distances 

 respectively for this second plane ; and accordingly 

 Mech. 31 



