180 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES — 
it is evident that this has no influence on the use here made of 
the auxiliary proposition. 
23. 
Drawing through each point of the surface anormal, and de- 
noting by p the distance of an indeterminate point upon it from 
the origin situated in the surface itself, the distance being con- 
sidered as positive on the inner side of the surface, the potential 
V of the masses may be regarded as a function of p and of two 
other variable quantities, which in some way or other distin- 
guish the several points of the surface from each other; and it 
is the same with the partial differential coefficient oe the 
value of which is to be taken into account here only for the — 
points which lie in the surface itself, or for p= 0. As this 
has exactly the same signification as P, if there are masses dis- 
tributed only in the internal space, or in external space, or in 
both, but none in the surface itself, then we have in this case 
“of as .ds=40M. 
dp 
In the contrary case of the whole mass being distributed solely 
in the surface itself, so that the element ds contains the mass 
kds, A and P have no longer the same signification ; it is 
manifest that the latter quantity is here in relation to p what 
X° is in relation to # in Art. 15; ~ on the other hand, has 
two different values, namely, P— 27k, and P + 27k, accord- 
ing as dp is regarded as positive or negative. Now, as kds 
is evidently equal to the whole mass M! distributed over the 
surface, and agreeably to the theorem of the preceding article 
j= ds = 22M’, we have either 
adv av 
ae = aed = ! 
is ds =0, fT ds=4nM!, 
: dV ‘ f 
according as we understand by ce the value which obtains 
everywhere without, or within the surface, and thus in the first 
case the integral is exactly the same as if the mass M! belonged - 
to the space without, and in the second case to the space within. 
