18 
MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
It will now be apparent that we can eliminate any two of the three unknowns, A, 
B, and C, from equations (67)-(69) by this method. The differential equation satisfied 
by the remaining unknown (say A) will be 
where, symbolica 
A = 
AA = 0. 
p 2 D 2 , p*D i A D 0 , D 0 
p 2 D 3 A D 2 , + d 2 , d 3 
D 2 , ipD 0 , ip D 0 + Do 
(7 
3), 
(74). 
We may expand this determinant according to the rules already laid down for the 
manipulation of the D’s, and so obtain 
A — ip° D 4 + a 4 D 6 A ^ 3 D 4 + ,P 3 D 6 A A D 4 . • • • (75). 
§ 18. We can now see the explanation of the difficulty which occurred in § 16. 
The occurrence of the term D G in A points to a differential equation of the sixth 
order, which is satisfied by any one of the quantities A, B, or C in the general case, 
in which p does not vanish. As soon, however, as p is put equal to zero, the 
expression for A reduces to D 4 , and the differential equation is one of the fourth order 
only. It therefore appears that by putting p — 0 before solving the differential 
equations, the order of these equations is reduced automatically, and two solutions 
are entirely lost from sight. 
These two last solutions, it is easy to see, are solutions which do not approximate 
to a definite limit, when p> approximates to zero. The remaining four solutions will 
approximate to the same forms as would be obtained by putting p = 0 before solving 
the differential equations. Thus, instead of equation (59), we must write the complete 
limiting solution for A in the form 
L' A — E lt /j (r) A E 3 f 2 (r) A E 3 / 3 ( r ) + E 4 / 4 (r) A E 5 / 5 (r. 
p =0 
_L It f la 
\ A- 
* I have not found it possible to investigate the form of these two last solutions in the general ease, but 
it is easy to examine the nature of the solutions at infinity, when the nebula extends to infinity, and this 
enables us to form some idea as to the general nature of the solutions. Suppose that at infinity we have 
j , 1 dTjr dX 
r = x Xp dr dr 
± a 2 r~* 
in which a is real, then it can be shown that A = (r, p) A', &c., in which A', B’, C', are functions of 
r only, and 
(r, p) = E 5 cos (2 Jan (n + 1 ) r~ si2 /isp) + E G sin (2 Jan (n + 1 )r~ sl2 /isp) 
when the negative sign is taken in the above ambiguity, the circular functions being replaced by hyperbolic 
functions when the positive sign is taken. The value of ip is wholly real when squares of ip may be 
neglected (c/. § 13). 
