36 
ON THE ORBIT AND PHENOMENA 
tlie chord of that arc, with a velocity equal to half the sum of the velocities at the 
two extremities of the arc, the specific gravity of the air being taken at the mean 
height of the chord above the surface of the earth. Representing now the anoma 
lies of the meteor at the two extremities of the arc by o and o', the velocities by 
v and v', the radii vectores by r and /, and the times of the meteor’s arriving at 
them by t and t’, the above equations will read 
Log. s' = %^q63/r+/_ _ 8956 \ and R _ pJ/v + *_ e (j _, ) 
Substituting for the factor t — t ’, in the latter equation, its value as given by the 
equation / 
r' 2 + r 2 — 2rr'cos(cd —o') 
bJvTV) ’ 
putting the angle included between the radius vector and tangent to the path = 0, 
and resolving R into its horizontal and vertical components, the expression for the 
former will read 
p s' fV + v'yr' 2 + r 2 — 2rr' cos (o • 
d s \ 2 / à (y + v') 
■o') „ 
-COS d, 
and for the latter 
p s' 
(V + v' 
d s 
\ 2 - 
-n sin 0. 
h ( v + v ') 
in which n represents the ratio in which the resistance in the vertical direction was 
increased by the increasing density of the air, as the meteor descended. 
Knowing from the elements the values of a, e, and o at the commencement of 
the disturbed part of the orbit, the values of r, v, and 0 at the end of the first small 
arc, if the orbit were undisturbed, were readily computed from the equations 
c 
rv 
r = 
a (1—e 2 ) 
a — r) 7«, 
1 4- e cos 
1(2 a- 
■ v = J —— and sin 0 = — in which latter, c represents the 
5 Cv / JO 
constant area described by the radius vector in a unit of time. Or, by substituting 
for c its value in terms of a , e, and 7i, the latter equation becomes 
0 =\l^~ e )h i To these values of r, v, and 0 corrections were applied for the 
sin 
rv 
resistance of the atmosphere in the horizontal and vertical directions, computed 
from the expressions for them given above, and from their values, thus corrected, 
new values of a , e, and o, were computed, with which to commence the next arc, 
the equations used for the purpose being as follows, viz :— 
hr_ 
2 h — rv 2 
a = ' 2 ’ c = tv sin 6 
5 e ~~ 
1 , and cos o 
ah 
a(l—e 2 )—r 
re 
Proceeding in the same way with the 2d arc, elements were found with which 
to commence the 3d arc, and so on from arc to arc till the whole disturbed portion 
of the .path was computed ; consisting, therefore, of a series of small hyperbolic 
arcs, each differing slightly in its elements from the one preceding. The value of 
n having never been investigated practically, so far as appears, and knowing no