156 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
wider sense, understand by the potential, that function of 2, y, z 
of which the partial differential coefficients represent the com- 
ponents of the resultant force. 
If we denote by p the whole force exerted at the point 2, y, 2, 
and by a, 8, y the angles which the direction of the force makes 
with the three coordinate axes, then the three components are 
Sd Pen ets dV LM, 
pooa=s 7, p gg? pcos y =& 77» 
~( Gavetag) * Coals 
If ds be the element of a line, either straight or curved, 
ae af = are the cosines of the angle which that ele- 
ment makes with the coordinate axes; and if § denote the 
angle between the direction of the element and that of the re- 
sultant force, then 
and 
S 
II 
dx dy dz 
cos = —— . cosa + 7. cos 6 + 7—» cosy. 
The force resolved in the direction of ds becomes consequently 
OV. d@ GN ng. ON see edV 
peosi==(‘ ne i dy ds ga 7s) = ds° 
If a surface pass through all the points in which the potential 
V has a constant value, it will, generally speaking, separate the 
portion of space in which V is greater than that value, from that 
in which it is less. If the line s lie in this surface, or be at least 
: : dV 
tangential to it at the element ds, then a Unless, then, 
the constituents of the whole force destroy each other at this 
point, or p = 0, in which case there can be no longer question 
of a direction of the force, cos § must necessarily = 0; whence 
we conclude that the direction of the resultant force at every 
point of such a surface must be normal to that surface, and will 
be directed towards that part of space where the higher values 
of V exist if «= +1, and to the opposite side ife=—1. We 
call such asurface a surface of equilibrium. As it may be drawn 
through any point, the line s, if not included in one surface of 
equilibrium, will at every point meet a different surface. If 
