20 SMITHSOXIAX MISCELLANEOUS COLLECTIONS VOL. 66 



For parabolic orbits we should have ri'- — 2 = o, thus a= co, accord- 

 ingly 



p = 2r sin- T, e=i, w=i8o° —2t, q= ^^ =rsm^ t, w'=i8o°, 

 (7 = 2t, Vo = o, 7r=i8o°+ O ±w. " (32) 



It can sometimes be of great interest to solve the problem in the 

 inverse order, that is to say, to determine the apparent radiant A', /8' 

 for a given day of the year when the sun's longitude is O and when, 

 therefore, we assume as given : 



(a) The necessary elements of the elliptic meteor orbit, or 



(b) The parabolic aphelion through / and b, or finally 



(c) The hyperbolic cosmic starting point, which may also be 



determined by / and b with Vo or v. 



Then according to the previous formulae a is given for the elliptic 

 orbit by the time of rotation, when it is not otherwise known directly 

 by the system of elements, for the hyperbolic orbit a is given either 

 by the equations (31) or (22). 



Further 



Since e^ and i are given among the elliptic elements, we can, there- 

 fore, also compute 



p = a(i-e'), smr=yt; (34) 



ri' 



and then we can also compute from the equations (24) the position 

 (A, /3) of the true radiant and from equation (20) A', j8', v' for the 

 apparent radiant. The zenith attraction can only be applied for a 

 definite latitude and siderial time. 



In hyperbolic orbits a is negative in the equation for v, but in 

 parabolic orbits it is assumed to be infinite. 



The system of equations (30) gives i and o- when we know O, 

 /, and b. Furthermore we obtain t from equation (29) , A and ^ from 

 equations (24) and finally, also, A', ^, v'. 



Equation (29) gives an associated pair of radiants for each initial 

 direction (/, b). Inversely, therefore, it always gives for each O two 

 associated values of (l, b) and with one position of (I, b) it gives the 

 associated apparent radiants for two far distant points on the apparent 

 celestial sphere. We usually find these two when we reflect that 

 cos(2t — 0-), even for a specified sign, always gives two values for 

 2t— 0-, therefore for a given value of o- there' are two values of t 

 corresponding to the two possible hyperbolas in each plane with a 



