104 Mr EARNSHAW ON THE NATURE 



shall act in the line of displacement. Hence those media which are 

 distinguished as uncrystallked, cannot consist of detached particles which 

 either attract or repel each other, with forces varying inversely as the 

 square of the distance; because it is assumed as a characteristic pro- 

 perty of such media, that the forces of restitution act always in the 

 direction of displacement. 



11. To find the force of restitution, when a particle is slightly dis- 

 turbed from its position of equilibrium. 



Let F, G', H' be the resolved parts of the force of restitution 

 parallel to the co-ordinate axis upon the particle at P; then F' is the 

 same function oi f + x, g + y, h + z, that F is off, g, h, and therefore 



F' = F + d f F.x + d 9 F.y + d h F.x + ... 



or, F' = d f V+ d}V.x + d f d g V .y + d f d h V .% + ... 



= d}V.x + terms involving a? 2 , y 2 , %-, xy, &c. 



because d f V = 0, d f d g V = 0, d f d h V = 0. (Art. 4). 



Similarly, G' = d\ V . y + ... 



and H' = d\V .% + ... 



Hence, if the system consisted of fixed particles, the particle P 

 only being moveable, the equations for P's motion would be 



dr t x = d) V . x \ 

 tt t y = dlV.y 



d\z = dlV.%\ 



very nearly. 



It is remarkable, that — — + — + — = 0. 



x y x 



12. From this investigation it appears, that the force of restitution 

 parallel to any one co-ordinate axis depends only upon its displacement 



