172 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
and the integral is decomposed into the two following, 
26 pe hd Pia aA «pt : 
fat we, 2 . acy h d P 3 
in both which the validity of the above-given mode of conclusion 
is self-evident. 
(a’ 
Lastly, it is manifest that the values of /! en are the same 
for all the three values of # within infinitely small differences. 
Hence it follows that Q! + %°, Q°, Q" — k®, differ infinitesi- 
mally; and the same will accordingly hold good of f (Q’ + k°) d 4, 
Sf Q°d 4, fa" — k) d 4, or of the quantities X’ + 2 x k°, X°, 
X" — 2k. 
This important proposition may also be expressed thus: The 
limiting value of X, with infinitely decreasing positive 2, is 
X° — 27k, and with infinitely decreasing negative x, it is X°+ 
22k, or X changes twice suddenly by — 2 x k®°, as # passes 
from a negative to a positive value; the first time as # reaches 
the value 0, the second time as it passes it. 
16. 
In the demonstration in the foregoing Article ithas been sup- 
posed that the intersections of the surface with planes passing 
through the first coordinate axis at P havea finite curvature : but 
our result will remain valid even if the curvature at P were infi- 
nitely great, with the exception of a single case. It follows 
from the supposition of the existence of a determinate ane 
plane to the surface at P, that for an infinitely small p, — ¢ must 
p 
itself be infinitely small; but the two values will be of the same 
order only if there is a finite radius of curvature ; with an in- 
finitely small radius of curvature, = will be of a lower order 
than eg. We will now show that our results preserve their vali- 
dity in the latter case also, provided only that the orders of the 
two latter quantities be comparable. 
‘ a 
Thus if we assume — to be of the same order as p#, where 
e z 
x i ae a 
» denotes a finite positive exponent, then aed 
F 
4 
oP a 
represents a 
J 
