271 
1907 - 8 .] The Problem of a Spherical Gaseous Nebula. 
values of q found by Mr Green (’2737, '1867, '1450, '0667)* * * § for the four finite 
values of k, 1*5, 2*5, 3, 4, and in the zero value of q for k = 5, the case 
described in § 29 below. In this case equation (24) has a solution in 
finite terms, which gives t= JS.x for infinitely small values of x, and there- 
fore makes q = 0, for x — 0. 
§ 28. Two interesting cases, k = 1, and k = 5, for each of which the 
differential equation (24) is soluble in finite terms, have been noticed, the 
former by Ritter, f the latter by Schuster. J Ritter’s case yields in reality 
Laplace’s celebrated law § of density for the earth’s interior, (sin nr/r), 
which Laplace suggested as a consequence of supposing the earth to be a 
liquid globe, having pressure increasing from the surface inwards in 
proportion to the augmentation of the square of the density. With Ritter, 
however, the value of n is taken equal to 7r/R, so as to make the density 
zero at the bounding surface (r = R). With Laplace, n is taken equal to 
1 7 r/R to fit terrestrial conditions, including a ratio of surface density to mean 
density which is approximately 1/2*5. The ratio of surface density to mean 
density given by Laplace’s law, with n = ^ 7 rfR, is in fact 1/2 4225, which is 
as near to 1/2*5 as our imperfect knowledge of the surface density of 
the earth requires. 
§ 29. For the case /c = 5, Schuster found a solution in finite terms, which 
with our present notation may be written as follows :■ — 
®5G) — 
x J 3 
*/(3x 2 + 1) 
. (27). 
This makes t = 1 at the centre (o-/r = x = oc). At very great distances from 
the centre, (x = 0), it makes 
t = x J3 = and 
. r 
Using (27) in (25), we find 
m _ (k + l)So- ^3 
E e 2 (3a? 2 +l) 3/2 
and if in this we put x — 0, we find 
M_(k+1)So- ^3 
E e 2 
* See Appendix to the present paper, Tables I. ... IV. 
t Wiedemann’ s Annalen , Bd. xi., 1880, p. 338. 
J Brit. Assoc. Report , 1883, p. 428. 
§ Mdcanique Cdleste , vol. v., livre xi., p. 49. 
• (28). 
(29); 
• (30), 
