Mi\ Christie on the Luminous Arch of the 



as possible the same time, 5 h 30 ra P. M. in both cases, there 

 can, I think, be no doubt that our observations were made on 

 the same luminous band in the same position. Having, through 

 the kindness of Mr. Faraday, been favoured with Mr. Harris's 

 interesting observations on the Aurora, I have computed the 

 height of this arch above the earth's surface, from these ob- 

 servations and my own ; and although there may be some 

 doubts with respect to the absolute height, as determined from 

 these, in consequence of the unfavourable relative positions of 

 the two places of observation, yet, as the limits which they 

 determine are very different from the height most recently as- 

 signed to similar phenomena, I do not hesitate to publish the 

 results. 



Assuming that the arch was formed by a band of light, of 

 great extent, in a line nearly at right angles to the meridian, 

 and parallel to the earth's surface, it is evident that, although 

 different portions of this band were actually observed, the abso- 

 lute height will be determined from the observed altitudes of 

 the highest points, and the arc between the parallels to this 

 band on the earth's surface, in the same manner as if the ob- 

 servations had been made on the same portion, from places in 

 a plane at right angles to the arch. 



Let then a, be the angles of elevation of the same point in 

 the arch, at two places A arid B, in a vertical plane passing 

 through that point ; <y the arc on the earth's surface between 

 A and B ; the angle contained by two lines drawn from 

 the point in the arch, the one to A, the other to the centre of 

 the earth : then if S = it ( + /3 + y), we shall have 



cote = cos/3 c 



cos a sin tf 



or if /3 is the greater of the two angles a, j8, and - = cos ^, 



COS cc 



cot = 



sn 



If r is the radius of the earth, and h the height of the arch 

 above its surface, 



A 



/ 



sm 



