7i.] PRINCIPLE OF KINETIC ENERGY. 39 



hence the direction of v, depends on the normal component F n . 

 If this component be zero, the curvature is zero ; i.e. the path 

 is rectilinear. 



69. Instead of resolving the resultant force F along the tan- 

 gent and normal, it is often more convenient to resolve it into 

 three components, Fcosa = X, Fcos/3 = Y, Fcosy = Z, parallel 

 to three fixed rectangular axes of co-ordinates Ox, Oy, Oz, to 

 which the whole motion is then referred. If x, y, z be the 

 co-ordinates of the particle m at the time /, the equations of 

 motion assume the form 



Thus, the curvilinear motion is replaced by three rectilinear 

 motions. 



70. If the components X, Y, Z were given as functions of 

 the time / alone, each of the three equations (2) could be inte- 

 grated separately. In general, however, these components will 

 be functions of the co-ordinates, and perhaps also of the veloc- 

 ity and time. No general rules can be given for integrating 

 the equations in this case. By combining the equations (2) in 

 such a way as to produce exact derivatives in the resulting 

 equation, it is sometimes possible to effect an integration. Two 

 methods of this kind have been indicated for the case of two 

 dimensions in a particular example in Part I., Art. 232. We 

 now proceed to study these principles of integration from a 

 more general point of view, and to point out the physical mean- 

 ing of the expressions involved. 



71. The Principle of Kinetic Energy. Let us combine the 

 equations of motion (2) by multiplying them by dx/dt, dy/dt, 

 dz/dt respectively, and then adding. The left-hand member of 

 the resulting equation will be the derivative with respect to / of 



