or 
dP 
—= — 2x {r° a + r° eP+Q)| 5 
ar 
Bor a fluid Q= £ and - 
‚2 12 12 2 eT) ! 
ee (5 zat +5 )e— (1-2) (err? + 2) rar 
Pre p Eng (12) 
+ 2xr* P = const. | 
In this case therefore we have a first integral. If S is only 
different from zero, when 7 < #, the same thing is the case with 
P and Q, whether P be equal to Q or not. For r > R (42) then 
becomes always a first integral, if we put P=0O. In this case 
we can get another first integral for 7 > &, by subtracting the third 
equation (10) from (11), viz.: 
rep(E +27 — =) = come oN, van ol ore 
Ee Le 
5. I have not succeeded in finding other first integrals of the 
system (10); in what follows we shall therefore content ourselves 
with the calculation of the approximation already found by Lorentz ; 
but we shall for this purpose start from the equations (10), and 
besides we shall not suppose ®,, to be constant. However intricate the 
way may be in which the different quantities £,, depend on each 
other and on the field, £,, can only depend on 7; hence we put 
Ess = O(r). 
We suppose the values of the other %’s only different from zero in 
consequence of the gravitation and therefore we may suppose these 
values to be zero in first approximation. We now think p,q, and s 
expanded in a series of powers of x, and the expansion broken off 
after the term of the first degree in x. We then find from (10), 
neglecting terms with x° etc, 
x 
A GRE LANS Po 
rd: re Ue en 
From the first two equations it follows, that 
rp? (p'+2q') =econst. and r? (v'—q') = const. 
As p’ and q’ must be infinite for = 0, the two constants appear 
to be zero, hence p’ = q’ =0 and p=q=-—1. No terms of the 
first order will occur, therefore, in p and q. Further 
