16 
MR. J. II. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
*‘{a + ^ i» £ ( ’' 3B) - i (, ' 3A) I = 7* c + £ (XT > B • ■ <G0) ’ 
w 
liile equation (30) may, with the help of (28), be written in the form 
V = f/r. 
in which £ is now defined by 
A dr <i 
- =X(C-BT)- - + 
r 
p civ n (n 1) 
d 
• ( 01 ), 
( 02 ). 
Substituting for V from equation (27), and treating the equation so formed in the 
manner explained in § 15, we find, as the equivalent of equation (61), 
(i.) A volume equation, analogous in form to (51), namely, 
dr* 
• (63). 
(ii.) Two boundary equations analogous in form to (52) and (53). . . (64), (65). 
Thus the equations found in § 11 may be replaced by 
(a) Thi •ee volume equations, namely, equations (60), (63), and (31). 
(f3) Six boundary equations, namely, equations (32), (33), (34), (35), (64), (65). 
We may conduct the elimination of B and C from the three equations (a) in a 
symbolic manner as follows :—- 
Let D„ be a symbol which is used to denote any linear differential operator of 
order n, the differentiations being with respect to r. The symbol has reference 
solely to the order of the highest differential coefficient which occurs, and must in no 
case have reference to any particular differential operator. Thus we write D ; , 
indiscriminately for every operator of* the form 
a»- 1 
f ,L ( r ) 3 ,.u /«-i ( r ) 3 ,.»-i + • • • 
The laws governing the manipulation of this symbol are as follows 
(i.) — I)„<jf> = ILc/j, 
(ii.) 1) „<J) + D= 1 )„(() (n > m), 
(hi-) = D w+ll </». 
