5O KINETICS OF A PARTICLE. [91. 



91. In the general case of three dimensions any two of the 

 equations of motion, 



can be combined by the method of Art. 87, and we find thus 



-- 



d^x d^z d ( dx dz\ ,. ^ . 



'Tf-^a-^ir**-*)- 1 *-** (2I) 



^fd^dx\ = ^ 

 dt\ dt dt) 



The expression xdyydx now represents the projection on 

 the ^/-plane of the infinitesimal sector described by the radius 

 vector of the particle during the time dt\ similarly, ydzzdy 

 and zdxxdz are the projections of the same sector on the 

 planes yz and zx, respectively. 



92. The right-hand members of the equations (21) are easily 

 seen to represent the moments of the resultant force about the 

 axes of x, y, z, if it be remembered that the moment of a force 

 with respect to an axis is the moment of its projection on a 

 plane perpendicular to the axis about the point of intersection 

 of the axis with the plane (Part II., Art. 213). If the moment 

 of the momentum mv of the particle be defined in the same 

 way, the quantities mydzjdt mzdy/dt, mzdx/dt mxdz/dt, 

 mxdy/dtmydx/dt are the moments of momentum, or, as they 

 are also called, the angular momenta, about the three axes of 

 co-ordinates Ox, Oy, Oz. 



As the axes are arbitrary, the equations (21) express the 

 statement that the moment of the resultant about any fixed axis 

 is equal to the time-rate of change of the angular momentum 

 about the same axis. 



