28 T U R N E K , 7"tf /rt/ Solar Eclipses. 



Case I. r '^la - J?, Z= - ^- say. 



This case bears the closer resemblance to that of Note 11. 

 The limit for is determined by the vanishing of the denomi- 

 nator. Putting 2aR - FP' ^p"^, the denominator vanishes when 



pcosd = ^. 



Let a denote this value of 6. Then the integral becomes 

 A' r s'mddB 



Vrj {p-cos~0 - q-)^ 



which is the same as that considered in Note II., with/ written 

 for i? and q for {r - R). The value is thus 



^log,tan(^+^ 



Since q^ = {r- of -{a- R)- and vanishes when r=2a- R, the 

 density becomes infinite at this distance, as it did in Note II. 

 for r=R. Outside this distance the density falls off in a 

 manner somewhat similar to that already tabulated for Note II. 

 It is scarcely necessary to give tables, which would have to be 

 made for different values of a, since the study of Case II. shews 

 us that the original supposition will not fit the facts. But if 

 tables are required, perhaps the quickest way of getting one for 

 any value of a would be to utilise Table I., keeping the values 

 of a the same and calculating the corresponding values of r 

 from the equation 



p cos a = q 



R{2a - R) cos^a = (^ - R){r - 2a - R) 



or 



(r — a)" = a^ - R{2a - R) sin'-a. 



For instance \{ a= 2R, we have 



r/^=2 + (4-3sin2a)i 

 = 2 +.(l + 3 C0S-a)5. 



