16 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
It will be noticed tliat u rises from zero to a maximum of about 1 ‘ 66 , falls to a 
minimum of about ‘82, and then rises to unity. 
Since -5 dyjdx^ = 1 when x-^ — 1 , we have = l/2'4087 = -4152. 
M. Ritter has ’4143 for this constant, which he calls m. 
It appears from the Table that the density at the centre is 102 -^ times as great as 
that where r — M. Ritter’s solution is intended to make that ratio exactly 100, 
but this solution shows that we ought to have started with a sliglitly different value 
of 7 / to obtain that result. 
In the general solution of the differential equation d'hjjdx^ == — eyjx^ the two 
arbitrary constants may be taken to be the values of y and dyjdx when x is infinite. 
Now, we have taken arbitrarily y=5'329 when x is infinite, and the physical 
conditions of the problem imply that dyjdx is zero when x is infinite. For if dyjdx 
had any positive or negative value different from zero, it would mean that at the 
centre there was a nucleus of infinitely small dimensions, but of finite positive or 
negative mass. Now, cq is that distance from the centre at which the density is 
1 / 102'45 of the central density ; hence, we may regard as the arbitrary constant of 
the solution. Whatever be the elasticity of the gas, we may always take as our unit 
of length that distance from the centre of the nebula at which the density has fallen 
to 1 / 102’45 of its central value. Hence, the above table gives the general solution of 
the problem, subject, however, to the condition that there is no central nucleus. 
If we view the nebula from a very great distance, appears very small, and thus 
the solution of the problem becomes y = log 2 a;^. It is easy to verify that this is a 
particular algebraic solution of the differential equation, as is pointed out by Ritter 
in his paper.I found this solution very useful in a preliminary consideration of the 
problem treated in this paper. 
The next point which w^e have to consider is the form which the solution will take, 
if, instead of taking as the unit of length, we take any other value. 
The density at any distance and the elasticity are to remain unchanged, but are to 
be referred to new constants. 
Thus, tv, r, remain unchanged, but are to be referred to M, p, jSr, a, instead of to 
Pi’ 
Now, since w remains unchanged, 
ey — ^ 
and, since remains unchanged, 
/->o o o ^ 
p « = Pi Pi«l • 
Also 
a a 
— — iXi * 
* I have made use of this solution in a paper in the ‘ Proceedings of the Royal Society,’ Dec. 3, 1883, 
and it has also been refeired to in a paper by Sir W. Thomson, ‘Phil. Mag.,’ vol. 23, p. 287. Sir W. 
Thomson’s paper covers much the same ground as some of JM.^Ritter’s eailier papers, but was written 
by him independently and in ignorance of them. 
