399 
into Poisson equations for infinitely weak fields, and so the solution 
of these equations may be reduced to the solution of Potsson equa- 
tions, if we content ourselves with successive approximations. We 
start namely with supposing that the g’sand y’s differ little from the 
values that they must have at infinity ; which comes to this that 
the squares and the products of the differences with those ‘values at 
infinity’ are neglected. Then we have to solve ten Poisson equations, 
and we find the differences multiplied by the factor x. Then anew 
correction is introduced, multiplied by the factor x’; this new cor- 
rection is likewise the solution of a Poisson equation, the second 
member of which has now, however, been calculated by the aid of 
the first correction. Going on thus indefinitely, the whole solution is 
obtained in the form of a power series in x. For the case of a spherical 
body, that can be considered as an incompressible fluid, H. A. Lorentz 
has calculated the field, neglecting terms which are multiplied by 
x° and higher powers of x. I have tried to follow the method 
used in this calculation, as I have understood it from oral commu- 
nications of Prof. Lorentz, in calculating the field of two spherical 
bodies at rest with respect to each other, which I hope to publish 
in a later communication. 
2. The calculation of the field of a single centre requires only 
that of three functions of the distance to the centre, which may be 
seen in the following way, given by Prof. Lorentz. 
Let the origin be chosen in the centre of the attracting sphere. 
It is clear that the g’s and y’s can only be functions of the distance 
nm toythée centre. Let: g,, = U, 91s = 9s, ==) and;.g,, ==20. in, a point 
P, lying on the a-axis. The field being supposed stationary, g,, = 
=o — 4 — =O — Fi — 9, and ds reversion. Of one, of the 
three coordinate axes can have no influence on ds’, also g,,, Gis; 
Jas» Jai» Js; and g;, are zero. Hence 
ds? = ude? + v (dy’+ dz’) + wdt? 
= v (dx? + dy’? +-dz’?) + (u—v) da? + wit’. 
In this expression dx? + dy?-+ dz*=dl represents the square of 
an element of length in the space (a, y, 2); dz’ is nothing but dr”. 
We can, therefore, also write 
de = vd? Se (u—v) dr? pir ate Po Se. ND) 
and this does not contain anything that refers to the particular 
situation of the point P. If we had, therefore, taken P on an auxi- 
liary axis ve’, ie. if we had taken P arbitrary, ds* still would have 
been given by (3). If x,y,z are the coordinates of P, then 
