1814.] On the Aurora Borealis. 429 



observe them. In Scotland they are considered as rather indicating 

 fair weather, though this is a prognostic which I ecu not pretend to 

 authenticate from my own observations, for I have seen every kind 

 of weather follow them. 



5. From the observations of Mr. Cavendish and Mr. Dalton, I 

 think there can be no doubt that the arched appearance of the 

 Aurora Borealis is merely an optical deception, and that in reality 

 it consists of a great number of straight cylinders parallel to each 

 other, and to the dipping needle at the place where they are seen. 

 Mr. Dalton, indeed, has given a mathematical demonstration of 

 this in his Meteorological Essays, p. 160, to which I beg leave to 

 refer such readers as have not considered the subject with the re- 

 quisite attention. 



6. The height of these beams above the surface of the earth is 

 much greater than that of most other meteorological appearances. 

 There are two ways of calculating that height : one by means of a 

 single observation, first explained by Mayer in the fourth volume of 

 the Petersburgh Acts. It requires only the knowledge of the 

 latitude of the place of observation, the apparent altitude of the 

 Aurora, and the distance of the limbs of the arch in the horizon. 

 Let P D p I of the annexed diagram be 

 the meridian of the place, P p the axis 

 of the earth, D 1 the diameter of the 

 equator, N the summit of the Aurora, 

 which is supposed to be situated in the 

 plane of the meridian. Let O be the 

 place of the observer, and H R the 

 common section of the meridian and 

 horizon. Now let us suppose P M = a, 

 sine of N O R = m, the co-sine of half 

 the distance of the limbs = g, the sine 

 of the whole distance = r, the sine of P O = co-sine of the 

 latitude = q, the sine of (90° -f the latitude — the arch of 

 apparent altitude) = p, and ON=y. Mayer has demonstrated 

 that 



2 m a g 2 q 1 



This formula is too tedious for common use ; but Kraft, by a 

 very ingenious transmutation, has reduced it to logarithms ; for if 

 the numerator and denominator of the fraction be divided by p* we 

 obtain 



e 1 q* 



v 1 



y = 2 m a *~ 



J 1 — 





Now the fractional part of this equation is nothing else than the 



