291 
1907-8.] The Problem of a Spherical Gaseous Nebula. 
§ 4. With a view to obtaining greater accuracy in the results for 
monatomic gases, Mr T. J. J. See has extended Homer Lane’s series (5) as 
far as the term containing x 20 ; and with the aid of additional terms, 
obtained by means of logarithmic differences of preceding terms, he has 
calculated the values of t, p, m, etc., at a place very close to the boundary of 
the gas. From this he has been able to find with great accuracy the radius 
of the spherical boundary, and the total mass of gas, corresponding to the 
Homer Lane Function ® V6 (x) (see § 17). These results are published in a 
paper entitled “ Researches on the physical constitution of the heavenly 
bodies” ( Astr . Nachr. No. 4053, Bd. 169, Nov. 1905). They were found 
after Table I., on page 299 of the present paper, had been completed by 
the entirely different method given below, and they are a confirmation of 
its usefulness. 
§ 5. Eight years after the publication of Homer Lane’s famous paper, 
the problem of the convective equilibrium of a spherical mass of gas under 
its own gravitation only was dealt with very fully by A. Ritter, in a series 
of papers entitled “ Untersuchungen fiber die hohe der Atmosphare und die 
Constitution gasfbrmiger Weltkorper,” published in Wiedemann s Annalen, 
1878-1882. Numerical solutions of equation (1) are given for the fol- 
lowing values of k, 1*5, 2, 2‘44, 3, 4, 5 ; these solutions being obtained 
wholly by a graphical process, similar to the process described in § 7 below. 
§ 6. Professor Schuster, in a short paper to the British Association at 
Southport in 1883, pointed out that it was possible to obtain solutions of 
equation (1) in finite terms in the two cases k = 1, and k = 5 (Jc = 2 and 
k = L2). For k = 1, the solution, in the present notation, is — 
®i(x) = x sin — 
x 
• G); 
a result which was first given by Ritter. For k = 5, the solution is 
x JS 
® 5 (x) = 
>/(3a?+l) 
( 8 ). 
§ 7. The method of obtaining numerical solutions of equation (1) which 
has been adopted throughout the present paper, is derived from that indicated 
by Lord Kelvin on page 291 of his paper to the Philosophical Magazine , 
1887, referred to in § 2 above. An arbitrary trial curve, t Q , fulfilling the initial 
conditions t — A ; ~=A'; at x — a\ is taken for t. 
dx 
From this curve, t 0 , a 
curve representing — A , or — -fit is obtained by direct calculation. One in- 
L cc*’ dx 2 J 
tegration performed on this calculated curve gives the means of drawing 
( ~ ; and the curve representing t v which is obtained by integrating is 
