1867.] 



Distribution of Matter, ^c. 



483 



absolutely vacuous, the matter previously within it having been distributed 

 outside it. It is known further that the mass of a particle may be 

 distributed over an]/ surface whatsoever enclosing the particle without 

 affecting the external attraction, and in this way we see at once that we 

 may leave any internal space we jplease, however excentrically situated, 

 wholly vacuous ; nor is it necessary in doing so to introduce an infinite 

 density, by distributing the whole mass previously within that space over 

 its surface, since that mass may be conceived to be divided into an infinite 

 number of infinitely small parts, which are respectively distributed over 

 an infinite number of surfaces surrounding the space in question. These 

 considerations, however, though they readily show that the internal 

 distribution may be widely different from any that is com.patible with the 

 hypothesis of primitive fluidit}^, do not lead to the general expression for 

 the internal density. Circumstances have recently recalled my attention 

 to the subject, and I can now indicate the mode of obtaining the general 

 expression required in the case of any given surface. 



Let the mass be referred to the rectangular axes of x\ y, z, and let p be 

 the density, V be the potential of the attraction. Then for any internal 

 point Y satisfies, as is well known, the partial diiferential equation 



^ + ^+&^=-^"''' 



or as it may be written for brevity VV=0. This equation may be ex- 

 tended to all space by imagining the body continued infinitely, but 

 having a density which is null outside the limits of the actual body ; and 

 by adopting this convention we need not trouble ourselves about those 

 limits. Conversely, if V be a continuously varying function of cc, y, z, 

 which vanishes at an infinite distance, .ind satisfies the partial differential 

 equation (1), V is the potential of the attraction of the mass whose density 

 at the point {x, y, z)h or, in other words, 



V= JJJ^. dx' dy' dz\ (2) 



where r is the distance between the points {x, y, z) and {x\ y , z), p the 

 density at (x', y, z'), and the limits are — oo to +oo, is the complete 

 integral of (1) subject to the condition that V shall vanish at an in- 

 finite distance. 



This may be proved in different ways ; most directly perhaps by taking 

 the expression for the potential (U suppose) which forms the right-hand 



member of (2), substituting for p its equivalent — ^ VV', V being the 



same f auction of x', y', z that V is of y, z, and transforming the integral 

 in the manner done by Green when we readily find U = V. 



* Essay on tlie Application of Mathematical Analysis to the Theories of Electricity 

 and Magnetism, Nottingham, 1828, Art. 3: or the reprint in Crelle's c'ournal, vol. xliv. 

 p. 3G0. 



