182 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES | 
at the different points of intersection, and « being = — | for the 
uneven points of intersection (the first, third, &c.), and= +1 for 
the even ones. Further, if we consider along this straight line 
the prismatic space, of which the rectangle dy dz is a sec- 
tion, dw.dy .dz being an element of it, then the integral 
2 
vs (e ~) + gail =) aT 
being extended throughout that part of T which falls within 
this prismatic space, = = « bs lid ,dy.dz. This prism separates 
out of the bounding surface ai or, generally, an even number 
of portions, and if each be denoted by d s, and the angle between 
the axis of a and the normal to ds directed inwards by &, then | 
dy.dz=-+ cos &.ds, the sign + being taken for the uneven, 
and — for the even points of intersection. Consequently the 
above integral will be 
dV 
lied ar cos § ds, 
where the summation applies to all the elements of surface 
which are met. Now if the whole space T be entirely resolved 
into such prismatic elements, then all the corresponding parts 
of the surface will exhaust these completely, and we shall have 
advV\? d7V dV 
i) + V ae) aT = — [VT 0st ds, 
the first integration being extended throughout the whole space 
T, and the second over the whole surface. Now it is evident 
that cos £ may be considered equal to the partial differential co- 
efficients i p having the signification established in Art. 23, 
and that 2 may be regarded as a function of p and two other 
variable quantities which distinguish the several points of the 
surface from each other, consequently 
dV — Vv dx 
Se a) tv 7) aT = {1% 79 
It is, moreover, gail that in the case of the surface 
_ shall have two different 
values, we should always understand here the value belonging” ‘ 
to the internal space. 
By conclusions exactly similar we find 
itself containing masses, so that 
