276 Trans. Acad. Sci. of St. Louis. 



By differentiation of (4) 



dM 



CT,R,- 





P dW 



P dR 



R^-'' /dP^ 



P^ \dRi 



(5) 



By geometry and after substitution from (3) and (1) 



d-R= """^ ^ = wm- 



(6) 



o^'o 



dM 



Equating these values of -—. in (5) and (6), and we have 



dR 



the differential equation for pressure as function of radius. 



d^P 2 — n dP 1/dPV 4.7r7cP^R^" 



dR^ 



+ 



R dR 



1 j dPy 4.7rJcP'R^'' _ 

 ~P\^dR) '^~CPTjR^~^' ^'^^ 



^"0 



The solution of this equation is 



P= (1 — n2) - 



"""O 



27^^•i^2(l+n) 



(8) 



By equations (3) and (1) the density of the gas is there- 

 fore 



3 = 



PR'' 



OT,R, 



n=(l-'^^) 



CTM.*" 



o-"'o 



27rA;P"+2 



(9) 



The mass of gas internal to the sphere of radius R is, 



iJf = 47r I R'^ddR = 2 ( 1 + n) — j^R^-^' ( 10) 



The weight of a gramme at the surface of the sphere is, 



l.^^ 0/1-1. N^^O^o" 



^= ^772 = 2 (1 + n) 



R' 



J^n+l 



