454 
In this sum the term produced by the first body does not change. 
It constitutes, together with 3p + oP a nearly invariable quantity 
of the first order. If we put 
vt 3p + oP — zon, = 
in the first body (and a similar expression in the others), we may 
consider 9’ to be the density ; in (13), (14), (15) we then may replace 
o by o'. Indicating now again @' by @, we find 
0B : A 
En >) = He — we jo, 
where the parentheses about > mean that in each, body the part, 
i 
A (oke DAE 
relating to the body itself, is to be omitted. 
The solution of (16) is 
_ OB , (e(p)ds a 
gl HB ix? P—2A-—}4 > Mn … (vee 
Suppose now the bodies to be spheres in case they rest (radii 
R,, R,,...,R,); as they are moving they will have a somewhat 
other form in consequence of the contraction in the direction of the 
motion, but the values furnishes by (14), (15) and the last term of 
(16) will vary in consequence of this only in terms of the third 
order; this variation we must neglect. Also in (13) we may do so 
for the calculation of the 3 which occurs in the third and the sixth 
term of (17). As to the second term of (17), however, we must con- 
sider the contraction in the proportion (1 — LS Dede Pnthag 
G 
| odS = 4am;, 
(7) 
we get from (13) 
m; 4% 
i= — (1 —3=2;"*) 
Vi 1) 
r; representing the distance from the ¢th body, and the dimensions 
of the bodye being small with respect to r;. Calling the coordinates 
#,,,,, henceforth «, y, z, so that x; Y y, Zi represent the components 
„of the velocity of the ¢-th body, and supposing (ai. y Ji 2) to be almost 
equal in each point of the ¢-th body, we obtain from (14) 
dE dl 
Seo gi ape por 
Representing the velocity of the i-th body by v;, we get from (15), 
as the dimensions of the body are small with respect to 7;, 
miv = 
i= Elen ld 
x2" 
i Ti 
