﻿Diurnal Rotation on the Upper Atmosphere. 667 



meteorites must have been continually destroying them, so 

 that it seems extremely doubtful if the history ot the earth 

 is now written in its atmosphere. 



Even the sign of the correction is doubtful. For if 

 external orbits are absent there will be a lag, whereas if 

 there are any remaining from a period of faster rotation the 

 motion will be greater than that of the earth. Under the 

 circumstances the most that can be done is to estimate 

 the effect when the orbits are supposed absent. For this 

 purpose we must consider what values of V, 0, (f> are to be 

 excluded in obtaining the various averages. If particles are 

 projected from r with all velocities in all directions, then 

 since the orbits are reversible, those values of V,0,c/> are to be 

 excluded which give rise to orbits never cutting r Q . Take 

 p./ijr as current coordinates in the plane of the orbit. Then 



-p 2 + p^=zY 2 -2ga 2 (^ - ~\ and p 2 f = rVs'm0, 

 whence 



The apses are given by making the right-hand side vanish. 

 2oa 



For elliptic orbits ~^ > Y 2 , and the apses are the roots of 



the equation 



{~~ -V 2 )p 2 -2 9 a 2 p + r 2 Y 2 sm 2 = 0. 



One of these roots is always greater than r. The condition 

 for the second to be less than r. is 



(^-^V-S^aVa + r 



:n s 6/<o. 



If a molecule is returned for every molecule which goes 

 to infinity with a hyperbolic orbit, the same condition will 

 hold lor these. If this is not assumed the solution would 

 involve one root of this quadratic and would be rather 

 complicated. The equation may best be understood geo- 

 metrically. Let uvw, the three components of velocity, be 

 represented as cartesian axes. Then the distribution of 

 velocities is 



at / > im \- t -hm(a- + o 2 +ti'--2n\osin \) 777 



JN I J e n auavaio 



for values of uviv inside the hyperboloid 



-'i^ 1 ., «, ., 2<7« 2 r' 2 , „ 



-'- — {ir + ir + ic-) — ■ + 2 (u 2 -f r) =0. 



