24 Turner, Total Solar Eclipses. 



The distances from the sun's centre are, in terms of the 

 sun's radius, the cosecants of these angles, or 1-26 and 2"6t, the 

 ratio of which is 2*07 (log = "3 16). The above table is then 

 modified as follows : — 



Value of lo + Ii- 



Law of density r„ 



At distance T '26, /0 + /1 =20704 

 „ „ 2-6i,/„ + /i = -9191 

 Ratio of brightnesses =2-25 

 Ratio as power of 2*07 = I'l 

 Power of 2*07 for density = 

 Difference i"i 



The difference 17 has been reduced to i "3 : and the law of 

 density approaches the inverse fifth power instead of the inverse 

 4th. For our purpose we may take the index as - 4'5. 



NOTE II. 



The velocity is given by the equation 



2 C" 2 



^\a r 



and thus although v increases with r it is not expressible as a 

 power of r. But we can represent it approximately by a power 

 of r within a limited region. Let J^ be the Sun's radius. The 

 velocity is zero when r=2«; and if this occurs at the Sun's 

 surface 2a = R ; otherwise 2a<^R, and r is always greater than 

 2a. Let us compare the velocities at distances r and 2r. The 

 ratio v'Ji^ is (r- a)/(r- 2a) which tends to unity when r is 

 large compared with a. It is less than 2 unless r is less than 

 3a. Thus 7'Jz\ is less than 2- except within a possible thin 

 shell close to the Sun. This shell does not exist at all if R'^^a, 

 that is, if the square of the velocity of projection be greater than 

 fji/R: and its maximum thickness is R/2, when the particles 

 start from rest at the surface. 



