28 Mr Potter on the Aurora Borealis. 



known quantities may be determined. In the course of the 

 elimination it is convenient to put I — 1 -f f, m = 1 -f e 2 , 

 to = 1 -f- g -2 , and p = 1 + eg, and we find the value of a for 

 the particular case of each arch to come out this simple ex- 



nf'l — n in J' Whlch tel ' S US tbat 



a has only one value for each position of the arch, or fulfilling 

 the same data. From this value of a we easily find those for 



(j* -l_ g2 _ . j, g 



c, a>, od and R, as we have c = a g, x — ■ and 



' ' ' a -J- e c 



w' = — — • With these we easily find the height 



of the arch. Taking as example e = trig. tang. 10°, and 

 J '= trig. tang. 40°, and^ making g in round numbers = trig, 

 tang. 50°, we bring out the height of such an arch to be 276.7 

 English miles- 



I am myself convinced that there is a very great distance 

 between the limits in height at which arches may exist, and 

 this will be seen by any one who notices the different apparent 

 curvatures which they take for the same angular elevation ; 

 but I have a very singular observation, which goes to show 

 that the same arch may descend nearer the earth. On the 

 evening of the 20th February 1830, about half-past 11 o'clock, 

 I saw that there was an arch of an aurora ; and, from a bright 

 mass beneath its eastern extremity, there arose several stream- 

 ers, which appeared to pass through the arch, and ascend to a 

 considerable height above it, say to 30° altitude. The air was 

 very cold and frosty, with a north-west wind and clear sky. It 

 was very clear underneath the arch, though at first there were 

 some small and very black clouds, which assisted me in deter- 

 mining the elevation. At half-past 11 the altitude of the un- 

 der edge was very near 15°, and the breadth of the arch 5°. 

 A little before 12 o'clock, I estimated the height again, and 

 called it only 12° to 15°, which made me suspect that I had 

 before called it too much. A little after 12 o'clock the arch 

 became very distinct and uniformly bright, and its under edge 

 along the whole part, which I could see, — namely, from about 

 the magnetic north-eastward, was quite distinctly defined, and 

 the light appeared to be glowing off uniformly along the whole 



