70 
Gaseous Stars 
[ch. Ill 
64. It is generally only possible to solve this equation by quadratures. 
There are, however, two exceptions. When k = 2, the equation assumes the 
simple linear form 
K d ( . dp n 
(641) 
which was discussed by Laplace. This equation is linear in p, and its general 
solution is easily found to be 
p = A 
sin (cr — e) 
•(64-2), 
where c 2 = 27r<y/K, and A and e are constants of integration. We are not at 
present interested in the most general solution either of the general equation 
(63*3) or of the simpler equation (641). For if e has any value other than 
zero in the solution (64*2), p runs up to an infinite value at the centre of the 
star. At the centre of an actual star, p must remain finite, reaching a maximum 
of the usual type at which dp/dr = 0. The solution expressed by equation 
(64*2) only satisfies this condition when e = 0, in which case it reduces to 
= A 
sin cr 
.(64-3). 
Again when /c = l'2, equation (63‘3) has a solution in finite terms, first 
given by Schuster*, for which dp/dr is zero at the centre, namely, 
P = P .( 1 + ^)" i '(64'4). 
For other values of k the equation can only be solved by quadratures. 
Starting at the centre, taking an arbitrary value of p and assuming that 
dp ¡dr = 0 , we find that p must steadily decrease as we pass outwards; as soon 
as p reaches a zero value we know that the star’s surface has been reached 
and at this point the quadrature stops. 
65. The solution of the general equation (63'3) by quadratures has been 
R 
very fully investigated by Emden in his book Gas Kugelnf. Put — © for 
the quantity we have denoted by K, R being the universal gas-constant, p 
the molecular weight of the substance of which the star is supposed to be 
formed, and m the mass of a molecule of molecular weight unity. The adiabatic 
relation (63‘2) between p and p now becomes 
p — — ©p* 
J mp 
The usual Boyle-Charles law is expressed by 
(65-1). 
II 
(65-2), 
* British Association Report, 1883, p. 428. 
t Teubner (Leipzig, 1907). See also Encyc. der Math. Wissen., vol. vi 2 b, part 2, reprinted 
in book form “Thermodynamik der Himmels Körper” (Leipzig and Berlin, 1926).