85.] PRINCIPLE OF KINETIC ENERGY. 45 



The normal distance PP" = dn between two equipotential 

 surfaces is therefore inversely proportional to the force F. 



It also appears that whenever the particle in its path returns 

 to the same equipotential. surface, the work done by F is zero, 

 .and hence, by (5), the velocity assumes the initial value V Q . 



84. Let a, /3, 7 be the direction cosines of F at any point 

 P ; X, ft, v those of any straight line s drawn through P ; and 

 let $ be the angle between /^and s, so that cos (f> = a 

 Then the projection of Fon s is 



or, since by (6) aF=d(7/dx, $F=dU/dy, yF= 



'~ dx ds dy ds dz ds~ ds 



i.e. the projection of the resultant force on any direction is the 

 derivative of the force-function with respect to that direction. 



This follows also from the equations (6), since the directions 

 of the axes are arbitrary. 



If s be taken tangent to the equipotential surface passing 

 through P, we have F s =dU/ds = o ; if it be taken normal to this 

 surface, we find F 8 = FdU/dn, which agrees with (11). 



85. The force-function U determines, as has been shown, a 

 system of equipotential surfaces U= const. Starting from a 

 point P on one of these surfaces, say U=c (Fig. 14), let us draw 

 through P the direction of the re- 

 sultant force, which is normal to 

 the surface U=c (Art. 82). Let 

 this direction intersect in P 1 the 

 next surface, L7=c'. At P 1 draw 

 the normal to U=c', and let it 

 intersect the next surface, U=c", 

 in P". Proceeding in this way, we 

 obtain a series of points P, P 1 , P", 

 JP'", ..., which in the limit will form a continuous curve whose 



