ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 195 
36. 
In the investigation hitherto the quantity U has been left in- 
definite ; its application to the case, for which the potential of a 
given mass has been taken for U, prepares the way for the fol- 
lowing important 
THEOREM.—For a given arbitrary distribution of mass D, 
which is restricted either entirely to the internal space, bounded 
by the closed surface S, or entirely to external space, we may 
substitute a distribution E solely on the surface itself, with this 
result, that the effect of E will be equal to the effect of D, at all 
points of external space for the first case, and at all points of 
the internal space for the second case. 
For this it is only requisite that as the potential of D is de- 
noted at any point of S by U, and the potential of E by V, 
V—U be = 0 through the whole surface for the first case, and 
that it be constant only for the second case. Of course — U will 
be the potential of a distribution D’, which is opposed to D (so that 
each molecule of mass is replaced by a contrary one), and thus 
V—U becomes the potential of the distribution D! and E sub- 
sisting at once; their effects will consequently destroy each 
other in the first case, in all external space; in the second, in 
- the whole internal space (Articles 27 and 25), or the effect of 
D and E will be equal in the spaces in question. Moreover, 
the whole mass in E will, for the first case, be equal to the mass 
in D, but for the second case will remain arbitrary. 
The theorem, which was announced in the Jntensitas vis 
Magnetice, p. 10, and which was referred to in different parts of 
the Allgemeine Theorie des Erdmagnetismus, now appears as a 
particular case of what has been here demonstrated. 
ay 
Although, as has been already noticed in Art. 35, the actual 
complete deduction of the distribution E presents, in most 
cases, insuperable difficulties, yet there is one case in which it 
can be effected with more facility, which deserves to be par- 
ticularly mentioned. It is when U is constant, so that S is a 
surface of equilibrium for the given system of mass D. It is 
easy to perceive that we need here only speak of the case in 
which D is taken in the internal space, and the total mass is 
not = 0, as otherwise there would be no effect at all there to be 
represented by a distribution of mass through S. 
