Prof. Stokes on the Internal Distribution of Matter, §-c. 237 



function of cc, y, z, which vanishes at an infinite distance, and satis- 

 fies the partial differential equation (1), V is the potential of the at- 

 traction of the mass whose density at the point (x, y } z) is p ; or, in 

 other words, 



dx dy' dz (2) 



=B 



(where r is the distance between the points (a, y, z) and (V, y', z'), 

 p the density at (x' y y', z'), and the limits are — oo to -f oo ) is the 

 complete integral of (1) subject to the condition that V shall vanish 

 at an infinite distance. 



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

 taking the expression for the potential (TJ suppose) which forms the 



right-hand member of (2), substituting for p its equivalent — — W, 



V being the same function of x, y', z that V is of x, y, z, and trans- 

 forming the integral in the manner done by Green*, when we readily 

 find U = V. 



Suppose now that we have a given closed surface S containing 

 within it all the attracting matter, and that the potential has a 

 given, in general variable, value V at the surface. For the portion 

 of space external to S, V is to be determined by the general equation 

 W = 0, subject to the conditions V=V at the surface, and V=0 

 at an infinite distance. We know that the problem of determining 



V under these circumstances admits of one and but one solution, 

 though it is only for a very limited number of forms of the surface S 

 that the solution can actually be effected. Conceive the problem, 



however, solved, and from the solution let the value of y— at the 



a V 



surface be found, v being measured outwards along the normal. 



Now complete V for infinite space by assigning to the space within 



S any arbitrary but continuous f function we please, subject to the 



two conditions, 1 st, that at the surface it is equal to the given function 



V ; 2ndly, that it gives for the value of -r- at the surface that already 



got from the solution of the problem referred to in this paragraph. 

 This of course may be done in an infinite number of ways, just as we 

 may in an infinite number of ways join two points in a plane by a 

 continuous curve starting from the two points respectively in given 



* Essay on the Application of Mathematical Analysis to the Theories of Elec- 

 tricity and Magnetism, Nottingham, 1828, Art. 3; or the reprint in Crelle's 

 Journal, vol. xliv. p. 360. 



t To avoid prolixity, I include in " continuous " the requirement that the dif- 

 ferential coefficients of the function, to any order required, shall vary continu- 

 ously. What that order may be it is perfectly easy in any case to see. We may 

 of course imagine distributions in which the density becomes infinite at one or 

 more points, lines, or surfaces, but so that a finite volume contains only a finite 

 mass. But such distributions may be regarded as limiting, and therefore par- 

 ticular, cases of a distribution in which the density is finite ; and therefore the 

 supposition that £ is finite does not in effect limit the generality of our results. 



