166 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
II. For the case of O situated outside of ¢, we have to take 
into consideration only those do, for which the straight line 
drawn through O and a point of dc really meets the space ¢ ; the 
number of the points O!, O”, O'", &c. will, in this case, always 
be an even number, and the angles fp’, W’, p'", &c. will be alter- 
nately acute and obtuse, thus ds! . cos W’ = — r? do, ds!' cos 
= — 72 do, ds" cos p"' = —r'? dco, &c. Now, as the inte- 
gration i os . dr must be performed from r = 7’ to r = 7", 
and then from 7 = 7" to r= r'¥, &c., we obtain 
: ! I} Ul 
dc = oe as ds! 
kK" cos pl k eos \ 
aa ie) . ds!" + xe. = = a) Ay, | 
and by the second integration carried through all the dc, 
k cos 
1 Ne be 2 ag = ire 
consequently 
dad? V av Ge Vn 6 
dat dy? +, eae 
ve 
Although it has been assumed in this demonstration that the 
density varies continuously in the whole space ¢, yet this con- 
dition is not necessary to the validity of our results, which 
merely requires that at the point O the density shall vary con- 
tinuously in every direction, or that O shall be situated in a 
space, however small, within which the conditions are satisfied. 
If we call the potential of the mass contained in ¢his space 
= V’, and the potential of the remaining mass situated beyond 
= V", we shall have the whole potential V = V' + V", and, as 
according to the preceding article 
2V! 2VI 2\! 
= TF + Gp = tek, 
2-H 2 2Il 
then 
GaN ae No a? V 
If, on the other hand, this condition fail at the point O, and 
= —4A a Ko. 
