194. GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES ~ 
The differential coefficient a has at the surface itself two 
different values, and the vaiue which applies to a negative dr, 
or to the inner side, surpasses that which applies to the outer 
side by 47m cos 6, m denoting the density at the place of inter- 
section, and @ the angle between 7 and the normal. (Art. 13, 
where ¢, A, k° correspond to 7,4,m in the present article.) 
These two values are found by differentiating with respect to r 
the two expressions for V which apply to the internal and ex- 
ternal space, and then putting 7 = R (1 + yz). The first is 
1 
ER (Bi + 2B" (1+ y2)+3 B"” (14+ y2)?4, &c.), 
and the second 
— (Ao (ty 22+ Al (Lt y+ A" (L+y2)* + Be.) 
Therefore, if we multiply the difference by R (1 + yz)*, we 
have 47m Roos é.(1+yz)?= 
A° (1+ y2)—? + Al (1+ y2)—* + A" (1 4+ 2)? 4+ &e. 
+B (l+yz)t+2 B! (14+ y2)?+ 3B" (14+ y2z)F + &e. 
If we substitute fer A® A’, &c., the values from I., and for 
Be, B, &c., the values from II., and if we neglect quantities of 
the order y*, we obtain 
4nmRcosé.(1t+yz)? =P°4+3P' 45 P" + 7 P" 4+, &e. 
we Y (a° ae a’ aie q'' Je ql! +, &c.) 
—tyz (P°+3P' + 5 P"4, &c.); 
consequently, as the two last series destroy each other within 
quantities of the order y’, 
Pes CS 2 
4a Roos 4 
whereby the problem is solved. Instead of (1+y 2)—?, we may 
.(P°+3 P45 Pl" +7 P" + &c.), 
write 1—3yz, and omit the divisor cos 6, since, generally 
speaking, 0 is of the order y, so that cos 6 differs from 1 only by 
a quantity of the order y*. 
For the case of a sphere, where y=0, we have rigorously 
1 
m= in (P°+3 Pp’ +5 Pp! + VA lt + &c.,) 
as P? + P’ + P"” + Pl + &c, is the development of U itself. 
