MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
43 
where oq is a completely determined function of r. Thus, as far as a , the solution is 
seen to be 
P = Po + « 0 T P i- 
We shall now show that, as far as a 3 , the solution is 
p = po -fi « 2 cr 02 ~h GoqPi -T croqP 2 .(120). 
The substitution of this in equation (118) leads to 
Po + + «o' 1 P 1 + cr<r 2 P 2 
= e'V ( 1 + f5b Pj + 1 Pf> p,y + . . . + -*?i P 2 + 
fC 
K 
K, 
■ + . ..j, 
K J 
where cf> l stands in the same relation to oq as does 6 X to P\- The right-hand member 
of this equation is equal to 
eYL +*;£ + *- + *. Pl + (f* + J f£) P . + 
' L K- K K \ K K~J J 
in which the unwritten terms are of degree at least equal to 3 in a. 
Neglecting a 3 the equation is satisfied if 
_ f 10r , 003 
Po ~ e K » °"o2 — do j i ^2 + — 
cr, 
Po0i 
_ Po02 I 1 Po0i~ 
cr.i — - ~r 3 "g - 
( 121 ). 
( 122 ), 
(12.3). 
These equations determine o - 02 and oq uniquely. 
It is obvious that this method is capable of indefinite extension, and that the 
general form of configuration in the series will be given by 
P = do + a 2 (T o2 + a 4 cr 04 + • • • + ( a <Xi + & 3 cr 13 + oUoq- -f . . .) P 2 
-j- (oreTo + cdcr 24 + . . . ) P 2 + (a J o-o + a 5 cr 3S -j- . . . ) P 3 -fi &c. . (124). 
§ 41. Let us examine in greater detail the solution as far as a 3 , this being given by 
equation (115). The important question, as will be seen later, is the determination 
of the sign of cr 2 . We therefore pass at once to the consideration of equation (123). 
Written out in full, this becomes 
cr, = 
47rpn fir , 
OK 1 
G 2 
