OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 739 



the quantities c^z 2 and fyz 2 , and we may write m for /n. We may also choose 

 our axes in the manner which is most convenient. We shall therefore make 

 the axis of z that round which the system, if it were rendered rigid, would 

 rotate with velocity w, and we shall suppose this axis to be vertical, as 

 otherwise a steady motion under the action of gravity could not exist, and we 

 shall denote the horizontal distance from this axis by r. 



We may now write for the density of the gas at the point (z, r) 



Sn-S 



p = p [l + (2l n (<>} )~* (ma>V 2mgz)^ 2 (105), 



where p g is the density at the origin. 



When n is a very large number and when the second term of the binomial 

 is very small compared with unity, we may write for this the exponential expression 



P = Pe ............................... (106). 



If m is the mass of a molecule of hydrogen, /zm will be the mass of a 

 molecule of the kind of gas considered, where p. is the chemical equivalent of 

 the gas. 



Also if T is the temperature on the centigrade scale, and a the coefficient 

 of dilatation of a perfect gas, then since the " velocity of mean square " of 

 agitation of the molecules of hydrogen at 0"C. is I'844xl0 5 centimetres per 

 second, the kinetic energy of agitation of a system contain ing n molecules of 

 any kind will be 



and the difference between this and the energy of internal motion may be 

 neglected. 



We thus find for the density at any point 



' (1-844) HO'" (1+aiO /1A'7\ 



P = P? ............................ (107). 



Let us now consider a tube of uniform section placed on a whirling table 

 so that one end, A, of the tube coincides with the axis while the other end, 

 B, revolves about the axis with the angular velocity w. The linear velocity 

 of B is car, and we shall suppose, for the sake of easy calculation, that this 

 velocity is one-tenth of the velocity of agitation of the molecules of hydrogen. 



932 



