418 H. A. Newton— Biela Meteors of Nov. 27th, 1885. 



quantity sought, namely, the angle through which the direction 

 of path has been changed. The asymptote of the hyperbola 

 makes then an angle z+x with the vertical. Let the velocities 

 at infinity, at the earth's surface, and what should be attained, 

 at the perigee of the hyperbolic orbit, be respectively v, v 1 and 

 v u ; let p, j9j and p be the perpendiculars upon the correspond- 

 ing tangents ; let g, a, b, and e, be the force of gravity, the 

 semi-axes and the eccentricity of the hyperbola; let d be the 

 angle at the earth's center between the line? to the observer 

 and to the center of the hyperbola, a the angle between the 

 original direction of the meteor and the conjugate axis of the 

 the hyperbola, and r the earth's radius. 

 Then we have, 



by conservation of areas, vp = vj) 1 =v p ; (1) 



by conservation of energy, v*— v i = 2gr, and p Q {v*— v' i ) = 2gr 2 ;(2) 



( a=b tan a, p=b, e=cosec a, 

 by nature of the hyperbola, < r(l+e cos 6)=a(e 2 — 1), (3) 



( p=p o (t&n a + sec a), (4) 



Also p=r sin z. 



From (4) ^- — ^-°=2 tan a, 



p Q p 



and from (1) and (2) 



p p = 2gr 

 p p pv'' 



qr 

 hence ^-j =p tan a=b tan a=a. (5) 



Again from (1) and (2) £= (i + ^)*-|-, 



hence b=p=rs,inz( l+-~ r =sinz(r 2 + 2ar)* (6) 



-, n , s r, ale 1 — 1) 1 a . ... 



and from (3) cos#=- =— cos ff-sm a. (7) 



\ i r e : e r 



But evidently d + z + x—-+a, (8) 



since each member is equal to the angle between the transverse 

 axis of the hyperbola and the asymptote. The equations (5), 

 (6), (7) and (8) enable us to compute a, b, a, d, and x for given 

 values of z. Taking #=32*12 feet, r=3956 miles, t>=9'9 miles, 

 we obtain this table of values of x with argument z. 



z 



X 



z 



X 



z 



X 



10° 



1° 0' 



40° 



4° 9' 



70° 



8° 0' 



20 



2 1 



50 



5 19 



80 



9 35 



30 



3 3 



60 



6 36 



90 



11 22 



Hence we may say that the earth's attraction changes the 

 radiant of the Biela meteors toward the vertical of the observer, 



