102 Mr EARNSHAW, ON THE NATURE 



4. That parametric surface which contains a point of neutral at- 

 traction will be a cone, which is asymptotic to all the hyperbolic para- 

 metric surfaces belonging to the other points. 



For, the parameter of the surface passing through K, the point of 

 neutral attraction is C", and therefore the equation of it is 



= d}V. a? + d 2 g V. f + d 2 h V.z\ 



which is the asymptote of the surfaces included in the equation, 



2(C- C) = d)V.x? + d* g V.f + d\V.%\ 



6. If the position of equilibrium be such, that only one of the 

 quantities d) V, d 2 g V, d\V is negative, as for instance, d)V; then the 

 axis of the asymptotic cone will coincide with the axis of x ; and all 

 points within this cone will have hyperboloids of one sheet for their 

 parametric surfaces, and their parameters will be less than C". The 

 points without this cone will have hyperboloids of two sheets for their 

 parametric surfaces, and their parameters will be greater than C 



If the position of equilibrium be such, that two of the quantities 

 d} V, cCgV, d\V are negative, as for instance, d\ V and d\ V, the axis 

 of the asymptotic cone and the parametric surfaces will be as in the 

 last case; but the parameters of points within the cone will be greater 

 than C", and of points without it, less than C. 



7. If the molecular forces are all repulsive, then the sign of V 

 will be changed : but the pai-ametric surfaces will be hyperboloids, as 

 before. 



8. If the position of equilibrium be such, that d) V = 0, d g V = 0, 

 and d\V = 0, then d f V, d g V, d h V, i.e. the attractions F, G, H, 

 being also evanescent, the particle is unattracted in every direction, at 

 least for small displacements from the position of equilibrium. An 

 example of this is afforded in the case of a particle placed within a 

 spherical or ellipsoidal surface, composed of attracting or repelling par- 



