30 Turner, Total Solar Eclipses. 



The limits are now 



e = o to « = '^, 



2 



since all directions of projection are represented in the shell of 

 radius i: Put 



p cos Q = s tan -i^ 

 — p sin QdQ = s sec-^\pd\p 

 (/>- cos-0 + /-)i = i' sec -d/. 



For limits, when = o, J tan ip =/>. Let this value of ^p be 

 denoted by \. 

 When 



TT 

 = -5 -i, = O. 

 2 



Hence 



-TIT- lo^„tan I - + - 

 f^rp °'' \4 2 



a similar expression to those obtained previously and it is most 

 easily tabulated by taking cot \ as the argument. We have to 

 find r from the equation 



j'-tan-A. =/" 

 or 



(r - a)-- = {a- R)- - {2a R - R^)co\:^\. 



Let us take the same particular value of a as in Case L, viz., 

 a = 2R. Then 



rjR = 2 + ( I - 3C0t"/\)j. 



Hence tan'-A. is greater than 3, or \ lies between 60° and 90°. 

 The former table only contains 5 values of \ (or «) between 

 tliese limits, but they will suffice to illustrate the solution to be 



