168 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
13. 
If we denote by & the density at the surface element ds; by 
a,b,c the coordinates of a point belonging to it; by r the di- 
stance of that point from a point O, the coordinates of which 
are a, y, z; and by V the potential of the ar contained in the 
surface at the point O, then is V = =f{"= — extended through 
the whole surface; and lastly, if we denote by X, Y, Z the inte- 
grals 
oes (a se ash f (d =» =, f- (c Bes =e 
it is true that X, Y, Z have thus the same signification as 
aWid ld dV 
dx’? dy’ dz 
but, strictly speaking,:this is no longer the case if O is a point 
of the surface itself; and the inequality is of different kinds, 
according to the nature of the angle formed by the normal to 
the surface and the coordinate axis which it encounters. It is 
obviously sufficient to give here the relation with respect to the 
first coordinate axis. 
I. If that angle = 0, then the integral X has a determinate 
——, so long as O is situated without the surface ; 
value at O; on the other hand, a has two different values, 
according as dz is regarded as positive or as negative. 
II. If the angle is a right angle, the expression for X does 
not admit a true integration (inasmuch as a remark similar to 
that in Art. 7. will hold good), while a has only one de- 
terminate value. 
III. If the angle is acute, X is as in the second, and a 
as in the first case. 
Other modifications are introduced if there be at O an inter- 
ruption of continuity in respect either of density or of curva- 
ture. However, our main object does not require us to treat in- 
detail such exceptional cases, which can only occur in particular 
lines or points; therefore, in pursuing the subject more closely, 
we shall assume that at the points in question there is a determi- 
nate finite density, and a determinate tangential plane. 
