44 



KINETICS OF A PARTICLE. 



[81. 



81. As the force-function U is a function of the co-ordinates 

 x, y, 2 alone, an equation of the form 



V=c, (9) 



where c is a constant, represents a surface which is the locus 

 of all points of space at which the force-function has the same 

 value c. By giving to c different values, a system of surfaces is 

 obtained, and these surfaces are called level, or equipotential, 

 surfaces. 



82. The values of the derivatives of U at any point P(x, y, 2) 

 are proportional to the direction-cosines of the normal to the 

 equipotential surface (9) at P. But, by (6), they are also pro- 

 portional to the direction-cosines of the resultant force -Fat this 

 point. It follows that the resultant force F at any point P is 

 always normal to the equipotential surface passing through P. 



If the equation of the equipotential surfaces be given, the 

 resultant force F at any point (x t y y 2) is readily found, both in 

 magnitude and direction, from its components (6) : 



fdU\* , /dA2 



-H-T +{-] ( I0 ) 



\ ox j \ oy J \ 02 J 



83. As the particle moves in its path from any point P to an 

 infinitely near point P 1 (Fig. 13), it passes from one equipoten- 

 tial surf ace (7=cto another U=c'. 

 Its velocity meets these surfaces 

 at a varying angle, while its ac- 

 celeration, which has the direc- 

 tion of the resultant force F, is 

 always normal to these surfaces. 

 The work done by F as the par- 

 ticle moves from P to P' is 



Fds cos (F, ds) = Fdn t 

 i, P" being the intersection of the 



where PP' = ds and />/>"=, 



normal at P with the equipotential surface passing through P 1 . 

 Hence, by Art. 72, 



(11) 



