100 Mh earnshaw, on the nature 



For, let f, g, h be the co-ordinates of the attracted body ; F, G, 

 H the attractions of the whole system upon it parallel to the co-ordi- 

 nate axes, then 



F = d f V,G = d g V, H = d h V. 



But the equation of the tangent plane at that point of the pro- 

 posed surface where the attracted body is placed, satisfies the differen- 

 tial equation, 



= d f F.df + d g V.dg + d k V.dh\ 



:. 0= F.df+G.dg +H.dh. 



This equation shews that the resolved part of the attractive force 

 is zero, in the direction of the tangent plane ; and therefore the whole 

 attraction is in the direction of the normal. 



2. For the sake of brevity, I shall denominate the surface V=-C, 

 the parametric surface passing through the point f, g, h. 



Different points in space may have corresponding different parametric 

 surfaces; any one may be found by assigning the proper value to C. 

 Their equations differ only in the value of the constant C, which, 

 for this reason, I shall call the parameter. 



If any parametric surface pass through an attracting particle, its 

 parameter will be infinite, because at that point V is infinite; in 

 which case the proposition will fail. The proposition is true of re- 

 pulsive forces, or if some of the particles exert repulsive and some 

 attractive forces ; but when the forces are all attractive, V can neither 

 be evanescent nor negative: since, however, it is infinite when the 

 attracted particle touches any one of the attracting particles, and is not 

 infinite in other positions, there must be some intermediate positions 

 which make V a minimum, and there may be positions in which V is 

 a maximum. 



3. The parametric surfaces which pass through points indefinitely 

 near to a point of neutral attraction, are in general similar concentric 



