A SWARM OF METEORITES, AND ON THEORIES OF COSMOGONY. 
13 
y = Aq A^{x — c) -{■ A^{x — cY , 
= ^1 + 2^3 {x — c) A- 3 Aq (x — cY A- ■ ■ ■ 
(20) 
In these series x may have any value, provided the series converges adequately. 
The convergence may be much improved by an artifice, ■which, however, I unfortu¬ 
nately did not discover until most of the computations were completed. Let us add 
and subtract log 2x^ on the right-hand side of (20). 
Now, 
log^ 2x^ = logv 2 c" A- 
2 lof 
1 + 
= log^ 2c“ 
+ 
X — c 
(x — c f , 5 , (x — cY 
.2 13 'j 
If, then, we write 
= Aq — log, 2c^ B^ = A^ — 
^2 — ^2 + 
2c2 
we have 
y = log,2x" A-B^A-— c) A-B.^{x — of + . . . , . . . (21) 
a more convergent seiies than that with the A's. 
The simplest way of computing the ^’s appears to be by first computing the A’s. 
The process for obtaining the numerical solution is then as follows :— 
We have the values of y, dylclx, ^cly-jdx^ when x —' 8, that is to say, of Aq, A^, A 2 
when c = 8. From these the successive N’s and B’s are computed, and the resulting 
series gives the values of y and dyjdx when a:; is 5 or r = '2. Starting from this 
point a new series gives the result when r = ‘3, another series gives the values for 
r = '4, and so on. Later in the calculation several values may be computed from one 
formula.* 
When the computation has been carried out to r = a, we have I’eached the 
spherical envelope, but that envelope may be replaced by another at any more remote 
distance from the centre. Thus, the integration may be pursued for values of x less 
than unity, and when the lower limit is zero the envelope is at infinity. 
If we write 
log u =z B^ A- B^{x — c) A- B. 2 {x — c)" A- ■ ■ ■ , 
* If the series be carried as far as Rg, several steps may be included in one series. For example, the 
first series, when c = 8, may be pushed even as far as r = '4 without serious error; for it gives 'j — 2‘960 
instead of the true value 2’965, and rlyfdx — '957, instead of the trno value '944. I l)ave not, however, 
been satisfied with tliis dc-gree of accurac_ 7 . 
