160 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
On the surface of the sphere both the expressions give the 
same value, and thus the potential varies continuously through- 
out space. 
For the differential coefficients we obtain, in the internal 
space, 
axa -4trke; ay == — Sky ~~ 
=Z=— Zakz; 
and in the external space, 
oe Yeas z= teks 
Here, also, the latter formule give the same values on the 
surface as the former ones ; thus X, Y, Z, also vary continuously 
throughout space. 
But it is different with the differential coefficients of these 
functions. In the inner space we have % 
dX Ad wie Siiurs 1A eA ae c. 
GoF WMS ude ae ad ee 
In the outer space, on the other hand, 
dX 4nkR3(3a%— 9?) dY_47kR®(3y?—p*?), dZ 
dx 3p? sail a 3 p° > dz 
_ 4ak R3 (3 2% —p*) 
ne oo 
On the surface the latter values do not coincide with the 
former, but surpass them respectively by the quantities 
4aka® Anxky? 42kz? 
Tape tase age 91S Peer 
Thus the above differential coefficients vary continuously in 
the whole of the internal and the whole of external space, but 
discontinuously in passing from one to the other; and in the 
separating surface itself they must be considered as having 
double values, according as dx, dy, dz are regarded as positive 
or as negative. 
It is the same with the remaining six differential coefficients, 
aX dX d¥ d¥ aZ az 
dq dz’ Adz’ ds’ d# ay 
which in the interior of the sphere are each = 0, and in their 
