8 ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
Hence equation (16) reduces to the form 
M Ul . n _ Ml f ST'\ , 3 / 3T'\ , JL f ^ 
M dt + C ° dt ” />o L 3^ V 0 &/ + (*° 3y / + 3* r° a? 
+ s(' / i) + iu'f HAH vU ■ < l8 >- 
a /, st ( 
dz \ c 
Since k 0 , T 0 are functions of r only, the bracket on the right-hand side of this last 
equation again reduces to 
a*o cl T-rorry , S/C cT n /r72 r T 
+ ar + ,cV ' i o 
0r 0r 1 ~° v A 1 dr dr 1 " v . 
and, cleared of accented symbols by the use of equation (2), this takes the form 
a*o ST, T7 r ’ r r i *afo i r7 or r 
a: a: + *o v ' T i + 757 -jT. + «i v ' T o 
— U 
or 01 
3 fdn n 0T, 
0 \ , yo ( cTn \ , a«o y;y 
' ° X Ur / + dr 0 
dr \ dr d? 
duj a_y o 0T o , . anj _ ar, 
" a.. 1 0 , v, ~r K o 77.9. r 
or [ dr dr "'° d/ 
Now equation (17) can be written in the form 
0 t~~2 „. 
K 0 
( 20 ). 
'N 
O,c o °t 0 
3r hr T ' 1,1 
+ k 0 V-’T 0 = 0. 
( 21 ), 
whence, by differentiation with respect to r. 
0 
d/c n 
3^ /cy 0 0 Tq\ , y 2 rp , —-o yorp _ ~ 
dr\dr dr I + u dr 0 + 0r V i o “ U - 
( 22 ). 
With the help of equation (22), the bracket in the second line of (20) reduces to 
2r 0 01V 
r~ dr ’ 
while, with the help of (21), that in the third line becomes 
9 is 0T 
_ ~__S) UJ -o # 
r or 
Again, if we substitute for k ] the value found for it in equation (6), the two last 
terms in the first line of (20) can be transformed as follows : 
a*i 3T n a / Tj \ 0 t„ t, ra* 0 8 t 0 
a7 a7 + ** v T » = *0 a- (wj e7 + Tt; 107 a7 + *0 V ' T « 
and the last bracket vanishes by equation (21). 
