The Problem of a Spherical Gaseous Nebula. 687 
symmetry with respect to y = excludes 8 of these. We 
may write 
U< 3 > = al* + blm* 
+ ex' I 2 + clx'm- + ei/lm 
+fx' 2 l + gx'y'm±hy' 2 L .... (33) 
and, by (14), 
x = Sal 2 + bm 2 + Icx'l + ey'm +fx' 2 + %' 2 . . (34) 
If the rays all proceed from the point x' = 0, y' = 0, the 
conditions for a well-formed primary focal line are 
a = 0, b = 0, (35) 
of which the first expresses that there is no aberration of 
this order for rays in the primary plane, i. e. } that the focal 
line is thirty while the second is the condition that the focal 
line is straight *. 
But if, while x' = 0, y' be left arbitrary, so that the source 
of light is linear, the evanescence of (34) requires, in addition 
to (35), that 
e=0, h = (36) 
Terling Place, "Witham, 
April 20, 1908. 
LXIII. The Problem of a Spherical Gaseous Nebula. 
By the late Lord Kelvin f . 
This paper was begun about the close of 1906, in order to fulfil 
a promise given at the end of the paper " On the Convective 
Equilibrium of a Gas under its own Gravitation only," published in 
the Philosophical Magazine, 1887 ; and part of it was commu- 
nicated by Lord Kelvin to the Royal Society of Edinburgh at its 
meeting on the 21st January 1907. Since then, however, important 
additions have been made to it, and the subject has been dealt 
with more fully than was originally intended. Unfortunately the 
manuscript was left incomplete at Lord Kelvin's death. It ended 
with § 35. 
However, from information which I received from Lord Kelvin 
while carrying out the earlier work connected with the paper, I have 
been able to write the sections from § 36 to the end. These complete 
all that Lord Kelvin desired to include in this communication ; 
* Compare Phil. Mag. vol. viii. p. 481, 1879 ; Scientific Papers, i. 
p. 410. 
f Communicated by Dr. J. T. Bottomley, F.P.S. 
