MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
7 
Let us write 
X = v ~ OT Vpo • • 
so that 
0y 0V' 1 0cfi ro' dp 0 
dr dr p 0 dr p 0 ° dr ’ 
then equation (12) becomes 
_i JL l ' _ / ?W\ 
dt 2 dr p 0 2 [f dr m dr ) ’ 
(13), 
and, by the use of equation (2), this is seen to be equivalent to 
d 2 u _dx . 1 / 3®o 0Po\ 
dt 2 ~ dr + Po 2 \ Pl dr ^ dr) ' * 
Equations (9) and (10) now take the simple forms, 
0 2 r 1 0^ 0% 1 0y 
0i 2 ?’ 2 d0 dt 2 r 2 sin 2 # 0c/> ’ 
(14). 
From these last two equations, we obtain at once 
1 0 
r 2 sin 0 d0 
+ 
0 2 
X 
r 2 sin 2 6 dcf r 
or, what is the same thing, 
02_ 
dt 2 
= V2 x - is 
dr 
7 * 
9% 
dr 
(15). 
§ 8. The equation of conduction of heat is, as far as the first order of small 
quantities, 
in which p, k, T stand for Po + p , k 0 -j- k , t 0 + t respectively. The notation is that 
of Kirchhoff ; the equation may either be written down from first principles, or 
regarded as a simplified form of Kirchhoff’s general equation.* 
Since there is thermal equilibrium in the undisturbed configuration, 
_0_ 
dx 
3T 0 \ 
0 dx J 
If 0T o\ 
0 2 r° 02 j 
= 0. 
(17). 
Kirchhoff, ‘Vorlesungen liber die Theorie der Warme, p. 118. 
