30 Trans. Acad. Sci. of St. Louis. 



It will not be necessary to insert a diagram illustrating 

 these functions, as the resulting curves are similar to those 

 found by Lane, and reproduced in Part I. And since the 

 temperature curves deduced from 



T=T 1 <T k ~ 1 (54) 



depends directly on that of the density, they will also be simi- 

 lar to those given by Lane. But it should be pointed out that, 

 while Lane's density curve depends on the value of k, and 

 in derivation is independent of the law of radiation, Hitter's 

 curve on the other hand depends on Poisson's law of radia- 

 tion, and is independent of the value of k. The fact that 

 the curves obtained by two such widely different processses 

 agree so closely, may be taken to show that the whole theory 

 of gaseous bodies despite its difficulty, is in a highly satisfac- 

 tory state. It is easy to see that the density is a function 



r 

 which follows a complex law, varying reciprocally as -^- 



To inquire into the theoretical nature of this curve, let us 

 express the density in units of the central density <r , as in 

 equation (B), and then we shall have 



R 



= 9{") < 55 > 



It is not easy to find the rigorous algebraic expression for 

 this function, but we may express it in Fourier's series as 



— J to be finite and continuous 



between r = 0, and r = R, and then put 



a — <p (x) = ^ b + b x cos x + b 2 cos 2x 



+ b 3 cos 3 x + . . . . + b m cos mx 



+ o 1 sin x + a 2 sin 2x 



+ a 3 sin 3x + .... + a m sin mx (56) 



