75-] PRINCIPLE OF KINETIC ENERGY. 4 ! 



co-ordinates x, y, z, of the velocity, i.e. of the time-derivatives 

 of x, y, z, and of the time t. If the motion of the particle were 

 completely known, that is, if we knew its position at every 

 instant, the co-ordinates would be known functions of the time, 

 say 



By differentiation the velocities v x = dxjdt, v y = dy/dt, v g =dz/dt 

 could be found ; and, substituting in (3), the integral would 



assume the form I <j>(t}dt y so that the work could be determined 



*AO 



by evaluating this integral. As, however, the motion of the 

 particle is generally not known beforehand, this motion being 

 just the thing to be determined, the integral cannot be evaluated 

 in the most general case. 



74. If the forces acting on the particle depend only on the posi- 

 tion of the particle, i.e. if X, Y, Z are functions of x, y, z alone, 



the integral I (Xdx + Ydy -f Zdz) can be determined whenever 



the path of the particle is given. For the equations of the path, 

 say 



* *) => fJi*> y> *) => 



make it possible to eliminate two of the three variables x, y y z 

 from under the integral sign, or to express all three in terms of 

 a fourth variable. In either case the function under the integral 

 sign becomes a function of a single variable, and the work of 

 the forces can be found. 



75. If the forces are such as to make the expression 

 Xdx -f Ydy + Zdz an exact differential, say dU, the integration 

 can evidently be performed without any knowledge of the path of 

 the particle between its initial and final positions. In this case 

 equation (3) becomes 



