192 GAUSS ON GENERAL PROPOSITIONS REGARDING FORGES 
cording to Art. 28, m'— m! would also = 0, or the two distribu- 
tions would be identical. 
Lastly, it must be noticed that there is always a distribution 
of mass in which the difference V — U has a given constant 
value. Let «denote an indefinite constant coefficient ; preser- 
ving for the first and third distribution the notation of the pre- 
ceding article, the potential of that distribution in which m = 
am° +p will be = « V°+ v, and the constant difference « V° + 
v — U may be given any desired value by a suitable determina- 
tion of the coefficient «. The total mass in this distribution is, 
then, no longer arbitrary, but = «M. It is evident, as before, 
that this condition of distribution also can be fulfilled in only 
one manner. 
35. 
The actual determination of the distribution of the mass on a 
given surface for each assigned form of U surpasses in most 
cases the power of analysis in its present state. The simplest 
case, where it is within our power, is that of the whole surface of 
sphere ; but we will forthwith treat the more general case, in 
which the surface deviates very little from that of a sphere, and 
quantities of a higher order than the difference itself may be 
neglected. 
Let R be the radius of the sphere, 7 the distance of any point 
in space from its centre, uw the angle between 7 and a fixed 
straight line, A the angle between the plane passing through this 
straight line and 7, and a fixed plane. Let the distance of an 
indefinite point in the given closed surface, from the centre of 
the sphere be = R (1+ y 2), where y isa very small constant 
factor, whose higher powers may be neglected, = as well as U 
being functions of wu and A. 
The potential V of the mass distributed through the surface 
will be expressed at any point of external space by a series de- 
ascending according to the powers of r, to which we give the 
form oe ; 
A° wk A'(=) +a"(=) + &e. 
r r r 
At any point of the internal space, on the other hand, by the 
ascending series 
0 ee " saps td m" ibd 
Bo + B+ Bi (a) +B ( g) + &: 
; 
4 
he 
t 
