[ 287 j 



XXXII. On the Equilibrium of a Gas under its own Gravita- 

 tion only. By Sir W. Thomson*. 



THIS problem, for the case of uniform temperature, was 

 first, I believe, proposed by Tait in the following 

 highly interesting question, set in the Ferguson Scholarship 

 Examination (Glasgow, October 2nd, 1885) : — " Assuming 

 Boyle's Law for all pressures, form the equation for the 

 equilibrium-density at any distance from the centre of a 

 spherical attracting mass, placed in an infinite space filled 

 originally with air, Find the special integral which depends 

 on a power of the distance from the centre of the sphere 

 alone." 



The answer (in examinational style !) is : — Choose units 

 properly ; we have 



dp _ 

 dr 



P 



j pr^dr 



r* 



(i) 



where p is the density at distance r from the centre. Assume 



p = Kr K (2). 



We find A =2, /€= — 2; and therefore 



P = i (3) 



satisfies the equation in the required form. 



Tait informs me that this question occurred to him while 

 writing for i Nature ' a review of Stokes's Lecture f on 

 Inferences from the Spectrum Analysis of the Lights of 

 Sun, Stars, Nebulae, and Comets ; and in the ' Proceedings 

 of the Edinburgh Mathematical Society ' he has given some 

 Transformations of the equation of Equilibrium. The same 

 statical problem has recently been forced on myself by con- 



* Communicated by the Author, having been read before the Koyal 

 Society of Edinburgh on the 7th and 21st February, 1887. 



Note of February 22, 1887. — Having yesterday sent a finally revised 

 proof of this paper for press, I have today received a letter from Prof. 

 Newcomb, calling my attention to a most important paper by Mr. J. 

 Homer Lane, " On the Theoretical Temperature of the Sun," published 

 in the American Journal of Science for July 1870, p. 57, in which pre- 

 cisely the same problem as that of my article is very powerfully dealt 

 with, mathematically and practically. It is impossible now, before going 

 to press, for me to do more than refer to Mr. Lane's paper ; but I hope to 

 profit by it very much in the continuation of my present work which I 

 intended, and still intend, to make. — W. T. 



t Lecture III. of Second Course of "Burnet Lectures,'' Aberdeen, 

 Dec. 1884 ; published, London, 1885 (Macmillan). 



