in Homogeneous Solid Bodies. 505 



it follows that as much heat must flow out from the interior 

 across the surface, as flows into the interior from the sources of 

 heat at the surface. Hence the total flux of heat from the original 

 surface to an adjacent isothermal surface in the interior is nothing. 

 Hence also the flux of heat from this latter surface to an adja- 

 cent isothermal surface in its interior must be nothing ; and so 

 on through the whole of the body within the original surface. 

 Hence the temperatui'e in the interior is constant, and equal to 

 Vp and therefore, for points at the surface, or within it, we have 



Now, if we suppose the surface to be covered with an attractive 

 medium, whose density at diff'erent points is proportional to p„ 



— -J- 11 ^- — - will be the attraction, in the direction of the 



axis of X, on a point whose rectangular coordinates are x, y, z. 

 Hence it follows that the attraction of this medium on a point 

 within the surface is nothing, and consequently /j, is proportional 

 to the intensity of electricity in a state of equilibrium on the 

 surface, the attraction of electricity in a state of equilibrium being 

 nothing on an interior point. Since at the surface the value of 



/ / - — — is constant, and since on that account its value within 



the surface is constant also, it follows, that if the attractive force on 



a point at the surface is perpendicular to the surface, the attraction 



on a point within the surface is nothing. Hence the sole condition 



of equilibrium of electricity distributed over the surface of a body 



is, that it must be so distributed that the attraction on a point 



at the surface oppositely electrified may be perpendicular to the 



surface. 



Since at any of the isothermal surfaces v is constant, it follows 



dv 



that T) where n is the length of a ciu've which cuts all the 



dn 



surfaces perpendicularly, measured from a fixed point to the 

 point attracted, is the total attraction on the latter point ; and 

 that this attraction is in a tangent to the curve n, or in a normal 

 to the isothermal surface passing through the point. For the 

 same reason, also, if p, represent a flux of heat, and not an elec- 

 trical intensity, 7- will be the total flux of heat at the variable 



extremity of n ; and the direction of this flux will be along n, or 

 perpendicular to the isothermal surface. Hence if a surface of 

 an infinite solid be retained at a constant tem]icrature, and if a 

 conducting body bounded by a similar surface be electrified, the 



