146 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



For a single particle the kind of application we can make of 

 our theory appears as an extension and generalisation of the 

 results of Chapter IV. ; the importance of such applications arises 

 from the theorem of Article 108, according to which the centre of 

 inertia of a body moves like a particle under the resultant of the 

 forces applied to the body. 



We shall not in what follows, as we have hitherto, continually 

 mention the frame of reference, and repeat that the motion 

 discussed is motion relative to the frame ; but this is always to be 

 understood. Thus when we speak of a fixed point or a fixed line 

 we shall mean a point or a line occupying a definite position 

 relative to the frame of reference ; when we speak of the path 

 of a particle we shall mean its path relative to the frame of 

 reference. 



161. Method of Particle Dynamics. The method of 

 formation of the equations of motion has been described in Article 

 80. It consists in equating the product of the mass of the particle 

 and its resolved acceleration in any direction to the resolved part 

 of the force acting upon it in that direction. The equations thus 

 arrived at are differential equations. The left-hand member of 

 any equation contains differential coefficients of geometrical 

 quantities with respect to the time. The right-hand member is, 

 in general, a given function of geometrical quantities. Although 

 there are many cases in which equations of this kind can be 

 solved, there exists no general method for solving them. 



Diversity can arise, in regard to the formation of the equations, 

 only from the choice of different directions in which to resolve. 

 Thus we may resolve parallel to the axes of reference, or we may 

 resolve along the radius vector from the origin to a particle, and 

 in directions at right angles thereto, or again we may resolve 

 along the tangent to the path of a particle and in directions at 

 right angles thereto. The most suitable directions to choose in 

 particular cases are determined by the circumstances. 



In regard to the solution of equations of motion we can only 

 premise that in cases where there is an equation of energy (Article 

 151), or an equation of conservation of linear momentum, or of 

 angular momentum (Articles 111 and 112), these equations are 

 first integrals of the equations of motion. 



