MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 17 
It must be particularly noticed that in general 
D ;i <£ — D„<£ = D „<f>. 
Corresponding, however, to any two specified operators of order n, say (D„), and 
(D„)o, it will always be possible to find two functions of r, say a and b, such that 
a (D„) 1 <f> — b (D„) 2 (f) = D, ; _,c£.(6G). 
In terms of this operator, the three equations (a) (p. 16) may be written in the 
following forms : 
P~ (DoA -j- D X B) + D 0 B + D 0 C = 0.(67), 
D 3 A -j- D 2 B + D 2 C + p 3 (D S A + D 2 B) — 0.(68), 
l P (D 0 B + D 0 C) + DoC -)- D 2 A — 0.(69). 
Now D„ is commutative with regard to functions of r, and is of course commutative 
with regard to p. This enables us to eliminate B and C from the above equations. 
To make this clearer, consider a simple case, say the pair of equations 
D 2 A = D ;i B.(70). 
D 1 A = p 2 B m B .(71). 
If we operate on (71) with d/dr, we get an equation of the form 
D.A = p 3 D m+1 B, 
and from this and equation (70), we can, with the help ot the property expressed in 
equation (66), deduce an equation of the form 
DjA = D„B + jAD„ ;+1 B. 
From this and equation (71) we can in a similar way obtain an equation of the form 
D 0 A — DJ3 + p~ D W+1 B. 
We may regard this as an equation giving A, and substitute for A in (71). In this 
way we obtain 
D„ +1 B -fi p 3 D, ft+2 B =0.(72), 
and the elimination of A has been effected. 
It will be clear that throughout this elimination we have followed a method which 
would have been successful in eliminating A if d/dr had been regarded as a mere 
multiplier. The result of the elimination is accordingly exactly the same as might 
have been obtained directly from the original equations (70) and (71), by regarding 
the D’s as multipliers and eliminating according to the ordinary laws of algebra. 
VOL. CXCIX.—A. 
D 
