176 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
of A being a right angle, so that the straight line only touches 
the surface, the two expressions coincide, and we have 
dV 
flee 0 
- ° cos B + Z cos C 
X* * * * 
As far as we have hitherto gone, the propositions contain no 
essential novelty; but it was necessary to state them afresh in 
connexion with, and preparatory to, the following investigation, 
in which a series of new theorems will be developed. 
19. 
Let V be the potential of a system of masses M', M!, M'.... 
at the points P’, P’, P"’....; v the potential of a second system 
of masses m', m!', ml" .... at the points p!, pl", pl”..... ; further, 
let V', V", V" be the values of V at the latter points, and 0’, v", v!” 
the values of v at the points P’, P’, P” ...., we then have the 
equation 
M! of M" ol Ml!" y+, &e. =a! Vm! V" 4m! V" +, &e., 
which may also be expressed by } Mv = 2 mV, if M repre- 
sents generally each mass of the first, and m each mass of the 
second system. In fact > My, as well as = m V, is no other than 
the aggregate of all the combinations see @ denoting the 
mutual distance of the points to which the masses M, m belong. 
If the masses of one or both systems are distributed, not in 
discontinuous points, but continuously on lines, surfaces, or 
solids, the above equation retains its validity, if, instead of the 
sum, the corresponding integral is substituted. 
If, for example, the second system of masses is distributed in 
a surface, so that the mass ds corresponds to the surface ele- 
ment ds, then 2 Mv = fs kV ds; or, when the same is the 
case with the first system, so that the surface element dS con- 
tains the mass Kd S, [Keds se EN. ds. It is important, in 
relation to the latter case, to remark that this equation still 
holds good if the two surfaces coincide ; for the sake of brevity, 
however, we will here only indicate the principal points of the 
mode in which this extension of the proposition can be rigor- 
ously justified. It is not difficult to show that these two in- ~ 
tegrals, in so far as they relate to one and the same surface, are — 
kira 
