40 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
The first term of’ our result, then, is exactly that resulting from the gaseous law, 
and the second subtractive term represents the action of the diminished velocity with 
which the meteorites move in the higher regions, when they are liberated from the 
equalising effects of continual impacts. 
By previous definition, ixMf{r) / a is the excess of the potential at radius a above 
its value at radius r ; hence. 
ivr'^ dr . dr. 
Now, since f{r) is only required for values of r greater than a, we may put lu equal 
to its mean value p, between the limits 0 and a. Thus, 
Hence, 
j ivr^ dr = j ivr^^ dr + ^pcd. 
Jo J 
n-r) = [£ ,1 1 w-'”' dr + = (l - “) + 3 f''”i t ch. 
If this form for f{r) were substituted in (43), we should obtain a very complicated 
ditfe3’ential equation for iv. We may, however, find two values of /{r) within which 
the truth must lie. 
First, if we neglect the attraction of all the matter lying outside of radius a, the 
second term vanishes, and we have, 
and the law of density is 
IV 
f{r) - 1 
a 
T 
(44) 
Secondly, we may suppose the density to go on diminishing according to the inverse 
square of the distance. We have seen in the preceding solution and tables that this 
is roughly the law of diminution for a long way outside the isothermal nucleus. 
According to this assumption, iv = Hence, in the second term of /'(r) we 
put IV = IVQ or/F = tvjz-. 
Hence, 
and 
f{r) = 1 
P IC „ , Wn ^ IV,, , . 
\ ~ z^-dz = — \ dz = -^{z — 1), 
h P P h P 
J-n 
jPF 
^pJ \ 
I a\ , Wr,, r , - 
The substitution of this value in (43) gives the la\v of density. 
In order to see the kind of results to wdiich these formulae lead,let us suppose that, when 
we have reached radius 2 in the adiabatic layer, collisions have become so rain as to. be 
negligible. Then the symbols in the formula! of this section have numerical values; and. 
