270 Proceedings of the Royal Society of Edinburgh. [Sess. 
the equation of the first and third members of (17), modified by (20) and 
(23), gives 
m _(k+ 1)So- dt /0 _ x 
E~ <? fa \ > 
§ 25. Let t = %(x) be any particular solution of this equation; we find 
as a general solution with one disposable constant C, 
i = C5[arCT«‘- l >] (26), 
which we may immediately verify by substitution in (24). Here 3(cc) may 
denote a solution for a gaseous atmosphere around a solid or liquid nucleus, 
or it may be the solution for a wholly gaseous globe, in which case 3-(x) 
will be finite, and 3'( x ) will he zero, when x = oo. Each solution $(x) must 
belong to one or other of two classes : — 
Class A : that in which the density increases continuously from the 
spherical boundary to a finite maximum at the centre. In this class we 
have dp/dr = 0 {dt/dr = 0), when r— 0; or, which amounts to the same, 
dpjdx — 0 (dt/dx — 0), when x — cc. 
Class B : that in which, in progress from the boundary inwards, we 
come to a place at which the density begins to diminish, or is infinite ; or 
that in which the density increases continuously to an infinite value at the 
centre. 
With units chosen to make 3(oo) = l, we shall denote the function J 
of class A by ©*, and call it Homer Lane’s Function; because he first 
used it, and expressed in terms of it all the features of a wholly gaseous 
spherical nebula in convective equilibrium, and calculated it for the cases, 
at = 1'5 and tc = 25 (k = If and 7^ = 1*4). He did not give tables of numbers, 
but he represented his solutions by curves.*' He did give some of his 
numbers for three points of each curve, and Mr Green, by very different 
methods of calculation, has found numbers for the case k = 2*5, agreeing 
with them to within X V th per cent. 
§ 26. By improvements which Mr Green has made on previous methods 
of calculation of Homer Lane’s Function, and which he describes in an 
Appendix to the present paper, he has calculated values of the function 
<h> k (os), and of its differential coefficient ®' K (x), which are shown in five 
tables corresponding to the following five values of k, 1*5, 2'5, 3, 4, oc. 
For the four finite values of k the practical range of each table is from 
x = q to x = oc, q denoting the value of x which makes t = 0. 
§ 27. There is such a value of x which is real in every case in which k is 
positive and less than 5. This we see exemplified in the four diminishing 
* American Journal of Science, July 1870, p. 69. 
