174 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
at 
rhb dh _h@@-a) da 
a ys Pe rs “db 
ec being considered as constant. The first integration gives upon 
this hypothesis, 
SEES tas fG2 Goa 
when the integrations extend rei the least é the wel 
value of 6 for every given value of c, and A*, r*, h¥*, r**, 
denote the values of and 7, corresponding to those limiting 
values. If, for the sake of brevity, we write 
RE EE e adh h(a—z)e da _ 
ree db eS 
Y= ftde+ ff abae, 
where the integration with respect to ¢ must be extended from 
the least value of this coordinate at the surface to its greatest 
value. In the double integral db dc represents the projec- 
tion of an indefinite element of the surface on the plane of 4, ¢, 
and therefore ¢ dg d 4 may be written for it; in which case 
y=/Tdc+ ff Ude.ds, 
where in the double integral we must integrate from eg = 0 to 
e =2', and from § = 0 to 6 = 2a. 
By conclusions similar to those in Art. 15, it is easy to per- 
ceive that the values of this expression differ by an infinitely 
small quantity, whether 2 be taken as=0, or as infinitely small ; 
or, in other words, the value of Y has one and the same limit, 
with positive and with negative infinitely decreasing values of 
a, and this limit is no other than the value of the above formula, 
if zis made = 0. By analogy, we will denote this value by Y°, 
in which, however, it must be remarked that we cannot call this 
then 
kbds ‘ ‘ : 
THE value of f° 3 for 2= 0 (inasmuch as this expression 
for z =O does not admit of a real integration), but only oNE_ 
value of that integral, namely, that value which is obtained if we 
integrate in the order which has been followed above. 
This result, moreover, requires (as in Art. 16) a restriction 
in the particular case, in which the radius of curvature at the 
point P is infinitely small, and likewise if = at this point is 
b 
oo ee ee 
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