q/'flr Shooting Star eclipsed in the Earth* s Shadow. '181 



be obtained by supposing the earth's shadow to be cylindrical 

 instead of conical ; the formulae then become extremely simple. 

 This is the same thing as supposing q = 0; we then have 



p2sin2(p + 2Rp(cos?-cosf cosZ)-R2cos2Z=0 . (8.) 



(9.) 



and, as before, using a subsidiary angle ^, but which is now 

 determined by the equation 



cot \I/ = cos 5 sec Z cosec f — cot <p, . . (10.) 

 we have 



p= cos Zcosec ftan— (11.) 



These two formulae furnish the means of calculating p in a 

 very few minutes with the aid of a table of logarithms and of 

 natural tangents. Such tables (Hutton's, for instance,) re- 

 quire to be opened seven times only. 



5. The formulae may be still further simplified by introdu- 

 cing, as one of the data, the angle contained between the great 

 circles which pass through the shooting star and the sun, and 

 the star and the zenith. As this angle may be directly ob- 

 tained from a celestial globe without calculation, it may be 

 worth while to exhibit the formulae with this substitution. I 

 believe, however, that although the formulae thus become more 

 simple in appearance, the calculation will be very little, if at 

 all, facilitated thereby. 



If we call this angle B, then by a well-known formula in 

 spherical trigonometry we have 



cos ?— cos <p cos Z= sin (p sin Z cos B ; 



and substituting this value in equation (8.), it becomes 



p2sin2^-t-2Rpsin(psinZcosB— cos2Z = 0. . (12.) 



Solving this equation, we have 



sin f 



P gnrz ^ ^^^^^ ^ "^ cot2 Z- cos B ; . (13.) 

 and using, as before, a subsidiary angle 4'j such that 



cotv|/=cosB.tan Z, (14.) 



we have 



/J = cos Z cosec (p tan — . . . » . (15.) 



