It may be verified that the second condition has been well chosen 
by integrating this expression (4) from O to op. Reducing the integral 
by partial integration, it will be fourtd that the condition (2) is 
accurately satisfied. In the critical point /” becomes 1, as found l.c, 
whence x—0. We have chosen the form of (4) in such a way 
0. Thence 
that in it we can easily take x 
Silas a7 
—— | s° f(s) ds + afl ds 
« 
0 r 
a solution which we might have written down directly from (3) 
after well-known theorems of the potential-theory. Now nearly all 
the simplifications whieh arise for x= 0, can also be used approxi- 
mately for small values of #. Besides, we must keep in mind that 
the equation (3) is also but an approximate one, which holds the 
more accurately, as will be seen by a closer consideration, as % is 
smaller. Therefore I used the following formula for the further 
calculations 
rl ao 
~ 
2 i 2 
g=— 77} en fot jds-+-— tsfds.-. = 2 ~ (5) 
e ‘ ‘ e . 
0 ve 
If we call d the radius of the sphere outside which f==0, then 
(5) gives for r > d 
It appears clearly from this form that the function g is sensible 
over a far greater distance than 7, and that outside the sphere of 
attraction the way in which y vanishes for increasing 7 no longer 
depends upon the course of 7. 
3. The considerations from the theory of electrons to be given 
now, may be understood as an extension of the treatment which 
Lorentz") gave for the scattering of light by an ideal gas. In the 
first place it appears that this treatment may be applied unchanged 
to our case, if it is allowed to consider separately a space with 
dimensions small with respect to the wave-length, surrounding a 
given molecule (and especially if we may assume the mean density 
in the space outside). According to the above calculation, this will 
be the case if the quantity z, and therefore also the distance from 
the critical point, is not too small. In the final formula of Lorentz 
1) H. A. Lorentz, These Proc. XIII, p. 92. 
98 
Proceedings Royal Acad. Amsterdam. Vol. X VIII. 
