OF THE MOTIONS OF A SOLID BODY. 239 



particles, multiplied by the squares of their 

 distances from that axis. 



Scholium 1. Consequently the rotatory inertia is 

 equal to the mass multiplied by the square of the radius 

 of gyration. This product is generally called on the con- 

 tinent the " momentum of inertia," but there is no reason 

 for abandoning the Newtonian acceptation of the word 

 momentum. 



Scholium 2. The elements and the squares of the 

 distances being always positive, the products must be al- 

 ways positive, and any addition to the bulk of a body, 

 wherever applied, will always increase the rotatory inertia. 



Scholium 3. The rotatory inertia will generally be 

 different with respect to different axes, but the various 

 cases are often easily deduced from each other, especially 

 when the axes are parallel.] 



342. Theorem. If ^,3/, and z be the co- 

 ordinates of the centre of gravity of a body, 

 of which the particles are subjected to the 

 forces P, Q5 and it, acting in the respective 

 directions, the sum of the quantities relating 

 to all the particles being denoted by the cha- 

 racteristic S, m being the mass, and Dm the 



particle, we shall have the equations m -^^7= 

 SPd/tx, m j^^SQ^m, and m j^=SRT>m.{A) 



The fluxional equations of the progressive and rotatory 

 motions of a solid body may easily be deduced from those 

 which have been demonstrated in the fifth chapter ; but 



