42 KINETICS OF A PARTICLE. [76. 



UQ being the value of 7 at the initial position, where v = v. As 

 most of the forces occurring in nature are of this character, this 

 particular case is of great importance, and deserves careful 

 study. 



76. The expression Xdx + Ydy + Zdz will be an exact differ- 

 ential whenever there exists a function U of the co-ordinates 

 x, y, z alone (i.e. not involving the time or the velocities ex- 

 plicitly), such that 



Z. (6) 



dx By dz 



If these conditions are fulfilled, we have evidently 



Xdx+ Ydy + Zdz = dU. 



The function 7 is called the force-function, and forces for which 

 a force-function exists are called conservative forces. 



Hence, if the forces acting on a particle are conservative, in 

 other words, if they have a force-function, the principle of work 

 gives a first integral of the equations of motion. 



77. The conditions (6) for the existence of a force-function U 

 can be put into a different analytical form which is frequently 

 useful. Differentiating the second of the equations (6) with 

 respect to z, the third with respect to y, we find 



dydz dz dzdy dy 



whence dY/dz = dZ/dy. If we proceed in a similar way with 

 the other equations (6), it appears that they can be replaced by 

 the following conditions : 



dz dy dx dz dy dx 



It is shown in works on the differential calculus and dif- 

 ferential equations that these equations (7), or the equations (6), 

 which are equivalent to them, are not only the sufficient, but 

 also the necessary, conditions that must be fulfilled to make 

 Xdx+ Ydy + Zdz an exact differential. 



