See — Temperature of the Sun and Ages of Stars a,nd Nebulae. 31 

 where 



H-7T 



b m = ^ | ?(P) cos mfidp 



■IT 



i r 77 



■*=;/ 



(57) 



- | y> (,2) sin m^ 



7T 



7r>af> — 7r, ^ (^9) denoting any known value of a = <p(x). 



Then since the researches of Lane and Ritter furnish a 

 numerically for given arguments of the radius, x will not ex- 

 ceed unity, and the multiple angles are, of course, to be taken 



P 



as multiples of radians, - (57°. 3), whereby denotes the multi- 

 ple of the angle, and q the number of parts into which the 

 radius is subdivided. 



Assuming the transformation here indicated, we may inquire 

 into the law of density when the radius has shrunk from the 

 loss of heat. 



The new density curve will obviously be given by an equa- 



(R'\ 

 tion of exactly the same form, a = <p I — r I. The internal dis- 

 tribution of temperature in the first case will be defined by 

 T= T 1 a k ~ 1 ; in the second case, by T = 2 T 1 V k_1 ; whence we 

 see that the laws of distribution of density and temperature 

 are the same after shrinkage has taken place as before. 



Some persons who do not fully understand the problem 

 under consideration, have claimed that the functions which 

 define the internal distribution of density and temperature 

 change with contraction ; that these functions depend upon 

 the linear dimensions of the globe rather than upon the prin- 

 ciple of convective equilibrium, and would give a new law of 

 temperature at each stage of the shrinkage. 



It will perhaps be evident from the preceding differential 

 equations or from the curves which they satisfy, that the 

 density and temperature are independent of the linear dimen- 

 sions of the gaseous globe. 



