of a Spherical Mass of Gas. 461 



The equation of equilibrium now becomes 



or 



dp Kp ( r 9 , 



. - S = — irri\ I pi-dr 



y-1 <//• J ^ 



(4) 



(5) 



Bv means of the substitution 









P 7 



-i 



1_ 



3/ 



~-> 





this las 



t equ 



ation reduces to 













dy 



4ttK(7 



-i) 





1 





dz~ 



C7 



« 



Now 



putt] 



ng, for 



convenience, 







(6) 





•V 4ttK(7-1) j 



and differentiating with respect to the upper limit, we get 



i 



da; 2 



(?) 



Being unable to find the general solution of this equation 



I have concentrated my attention on a single solution of the 



type which applies to the physical problem, and I shall shortly 



show that this single solution can be adapted to any mass of 



gas. 7 



Let us take 7 = 1*4. If we start with y = when ~-z=m 

 ' J dx 



and x = a, m and a being both positive, then so long as the 



curve for x and y does not cross the axis of :c again we know 



that y<mx ; consequently 



d*y m 2 ' 5 x 2 ' 5 



therefore m ^< 



dx 





The right hand side of this inequality is finite when the 

 upper limit is infinite. Hence, if we start with suitable 



