170 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
0 ! 
clear that the mutual relation between X° — X! and aes = pole 
dx dx 
a 
ris exactly the same as between X° and 
i.e. between X and 2 
on : whence the propositions in the preceding Article follow of 
themselves. 
is: 
For the more general investigation, it is advantageous to take 
the origin of coordinates at a point P, situated in the surface 
itself, and to make the first coordinate axis perpendicular to the 
tangent plane at P. If we denote by the angle between the 
normal to the indefinite surface element ds and the first coor- 
dinate axis, then cos f ds is the projection of ds upon the 
plane of 6 and c; and if we call VW (6? + c*) = p, b = p cos 4, 
c = psin#, then p dp. dé will represent an indeterminate element 
of this plane, and the corresponding surface element ds = 
pdp.dd 
; the element of mass contained therein will be = 
cos 
k 
cos 
We will now examine how far the value of X changes dis- 
continuously, as the point O in the first coordinate axis passes 
from one side of the surface to the other, or as 2 passes from a 
negative to a positive value. In this question, it is obviously 
indifferent, whether we take into consideration the whole surface, 
or only an indefinitely small part of it enclosing the point P, as 
the share contributed by the remaining part of the surface to the 
value of X changes continuously. We may therefore take e 
only from 0 to an indefinitely small limitary value ,/, and 
hpdp.d 4, if for the sake of brevity, we write, / for 
suppose that in the surface so bounded hf and z vary every- 
where continuously. If for any determinate value of 4 we call Q 
the value of the integral if Mee Aa! taken from g = O to © 
p = ¢', then X = f' Q d 4; where the integration is to be taken 
from 6 = 0 to §= 27. ti 
We have now to compare the values of X for 2 = 0, for an in 
finitely small positive 2, and for an infinitely small negative il 
(the other two coordinates being always assumed = 0); we willy 
“tet aoree 
bi 
