26 Turner, Total Solar Eclipses. 



where K is a constant depending on the number of particles 

 ejected in unit time. To get the total density we must integrate 

 this expression between limits. The lower limit is clearly = o. 

 For the upper, we must include all directions of projection 

 providing particles which reach the shell ; some do not. The 

 direction for which a particle just reaches the shell is given by 



r- R 

 cos Q = = cosf(, say 



and for larger values of d than this the denominator of the 

 expression to be integrated becomes imaginary. 

 For the integration, put 



R cos Q = {r - R) sec -^ 

 - R sin Bdd = (r - R) sin \p sec'^\Ld\p 

 [R-cos-0 - (r - R-]i = {r- R) tan yh 



and we get, since R cos a = {r — R), 



K 



A / 



777,-log,, tan \ - + 

 VRr => V4 2 



-los,ntan 



VRr 



where R cos a = r - R^ and A" is a new constant. 



This becomes infinite at the sun's surface, i.e., when r=R: 

 which is otherwise obvious since a number of particles remain 

 at this distance for an indefinite time as close satellites. For 

 distances greater than this we can quickly calculate 



log]„tan( - + -\ = L 



say, as in Table I. If L varies at any point as r~", then 



logZ + n\ogr= const., 



and hence differences of logZ, divided by differences of logr, 

 will give n, as in the last column of Table I. The average value 

 of n is about 3 up to distance i"8, where il rapidly increases. 



