484 



Prof. G. G. Stokes on the Internal [May 16, 



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 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=Vq 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 ^ 



at the surface be found, y being miCasured outwards along the normal. 

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

 arbitrary but continuous* function we please, subject to the two conditions, 

 1st, that at the surface it is equal to the given function ; 2ndly, that it 

 dY 



gives for the value of— 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 directions, which curve may be either expressed 

 by some algebraical or transcendental equation, or conceived as drawn 

 libera manu, and thought of independently of any idea of algebraical ex- 

 pression. The function V having been thus assigned to the space internal 

 to S, the equation (1) gives, according to what we have seen, the most 

 general expression for the density of the internal matter. 



There is, however, no distinction made in this between positive and 

 negative matter, and if we wish to avoid introducing negative matter we 

 must restrict the function V for the space internal to S to satisfy the 

 imparity 



d?-^-d^^-^-d^->^' 



It is easy from the general expression to show, what is already known, 

 that the matter may be distributed in an infinitely thin, and consequently 

 infinitely dense stratum over the surface S, and that such a distribution is 

 determinate. 



We know that there exists one and but one continuous function apph ing 



to the space within S which satisfies the equation VV=0, and is equal to 



* To avoid prolixity, I include in " continuous " the requirement that the differential 

 coefficients of the function, to any order required, shall vary continuously. What tliat 

 order may be it is perfectly easy in any case to see. We may of course imagine distri- 

 butions 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 particular, cases of a distribution in which the 

 density is finite ; and therefore the supposition that j is finite, does not in effect limit the 

 generality of our results. 



