36 ON THE ORBIT AND PHENOMENA 



the 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 w', the velocities by 

 V and v', the radii vectores by r and r', and the times of the meteor's arriving at 

 them by t and t', the above equations will read 



, , log. .24763/?-+r' oncc\ j d p s' /v + v' \^.. ,,. 

 Log. s = _^__(_^. -3906), and B = %-(^-; (i-t )■ 



Substituting for the factor t — t', in the latter equation, its value as given by the 

 equation 



, ,, _r'^ + r^ — 2>t' cos (o — u') 



H^ + V') ' 



putting the angle included between the radius vector and tangent to the path = 0, 

 and resolving M into its horizontal and vertical components, the expression for the 

 former wUl read 



p s' /v + v' \}r' ^ + r^ — 2rr' cos (o — o' ) a 

 dJ\2~) i{v + v^ ^^^ ' 



and for the latter 



2^ s' /v + v'\^/^ + r"- — 2rr' cos (o — co') . n 

 ds\ 2 / !(« + «') 



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 co at the commencement of 

 the disturbed part of the orbit, the values of r, v, and d at the end of the first small 

 arc, if the orbit were-undisturbed, were readily computed from the equations 



a(l— e') 1(2 a—r) h, , . ^ c 

 *■ ^ 1+ecos cj, ^ ^ \~ar ^^" ^ ^^ "^ ^^^^^ ^^**^^"' ^ represents the 



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 h, the latter equation becomes 



sin 6 =\^^^i-^^ ^ '" To these values of r, v, and 6 corrections were applied for the 



TV 



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 : — 



7ir 



2h — rv' 



• A 1 c- , a(l — e^) — r 



, c = rv sm d, e = ^ 1 —, and cos o = — ^^ 



N a/i 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 



