162 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
I f(a +e, b,¢e)(a—a)dt (3.) 
J ((G—2F + Oy FC=ae 
where the integral (1.) extended through the whole space ¢ is 
Vv 
the value = or X, at the point O. The integral (2.), in like 
manner, extended through the whole of ¢, is the value of aes 
at the point of which the coordinates are x + e, y, 2; we will 
denote this value by X + § It is manifest that this integral 
and the integral (3.) extended through the whole space 7’ are 
perfectly identical. So if 
the integral (1.) extended through # be . . . . J 
oe * through §,..be Jiianweeme 
. (3.) pn through 7° be; \\s shasta pane 
‘S through 6! be...) 4; tape 
39 
then X=/J+a,X+F=l 44. 
If we write f (a + e, b,c) —f (a, b,c) = A k,then the integral 
Ak 
B Fane ites Pe en 
((a— 2)? + (6 — y)? + (ec —38)")3 
U 
extended over 7° will be = est 
The results hitherto obtained hold good generally for every 
position of O. In the further development, the case in which O 
is situated in the surface itself will be excluded, or O will be 
assumed to be at a finite distance from the surface, either within 
or without ¢. 
Now if we make e infinitely small, the spaces 6 é! are two in- 
finitely thin strata of space at the surface of ¢. If we resolve 
this surface into elements d s, and denote by « the angle which 
a perpendicular raised outwards at d s forms with the first co- 
ordinate, it is manifest that « will be acute everywhere where the 
surface of ¢ adjoins 4, and, on the contrary, obtuse wherever it 
adjoins 4’. The elements of @ may thus be expressed by e cos ads, 
and the elements of 6’ by — e cos ads, whence it may easily 
Ave, re 
be concluded that passes into the integral 
ag ADs 2, 6) (6 eS 8 5 
KG a) st (by) he 8) ) a 
or, which is the same thing, into the imtegral 
k (a — x) cosa.ds 
cs PP 
