ON THE PERIODICITY OF THE AURORA BOREALIS. 



217 



In 1843 Eklof 1 published at Helsingfors, Russia, a mathematical discussion of the an- 

 nual changes of the auroral phenomena, making use of the same periodic function as I 

 have adopted. 2 



P m — 8.33 + u' sin. (m 30° + W) + u" sin. (m 60° + U") + u'" sin. (m 90° + V"). 



He also has reduced all the observations to the uniform standard of one hundred for 

 the twelve months of the year. As in his formula (m) is equal to zero for January 1, the 

 time counts from January 1, -whereas I take January 15 as the initial day of the year. 

 Accordingly my values of c x exceed by 15° and multiples of 15° the values of JJ X com- 

 puted by Eklof. This should be remembered in comparing his formulas with those which 

 I have computed for the same places. There is an advantage in this comparison, inas- 

 much as the observations which he used generally refer to a different period of years from 

 that which I included in my calculations. For this purpose I append his conclusions. 



Berlin, P„,=8.33-|-3.46sin.(m30 o -f 110° 22')+6.97sin. (;«60°+265° 27')+3.37sin. (m 90°+277°35'). 



London, P m =8.33+5.G6 sin. (m 30°+121° 32') -f5.33 sin. (m 60°+257° 55')+1.57 sin. (m 90 o +243° 26') • 



Upsal, P,„=8.33+5.42 sin. (m 30°+101° 48')+3.94 sin. (m 60°-f 283° 24') +0.94 sin. (m 90°+281° 220- 



St.Petersb'g, P,„=8.33+3.99 sin. (m30°+129° 41')-f 7.77 sin. (m 60°+278° 28')+1.17sin. (w90°-|-272° 30'). 

 Italy, P M =8.33-f-1.97 sin. (m 30°+ 48° 38')+4.85 sin. (m 60°-f 305° 49')+3.24 sin. (m 90°+245° 37')- 



Utrecht, P,„=8.33+2.29 sin. (m 30°+336° 8')+3.64 sin. (m 60°+251° 16')+1.40 sin. (m 90°+ 86° 34'). 

 Carlsruhe, P m =8.33+2.73 sin. (m 30°+337°28')+5.11sin. (m 60°+282°59')-|-2.34 sin. (m 90°+170° 0'). 

 Mt'morenci, P„=8.33-(-1.21 sin. (m 30°-f 71° 44')+4.07 sin. (m 60°-4-279° 46')+0.71 sin. (m 90°+14° 4'). 

 Wittemberg, P m =8.33+3.29 sin. (m 30°+104° 46')+5.33 sin. (m 60°4-303° 25')+0.76sin. (m 90°+300° 58'). 



i Acta Soc. Sci. Fennicse, II. 302. 



2 This is the same as I employed in 1845, in discussing the daily changes of temperature and magnetic declination. — Memoirs 

 of the American Academy, II. 44. Moigno, Lecons de Calculus, I. 527. Liouyille, Journ. de Math., II. 193. 

 8 As given by Bertholon, KiLmtz, and Gehler. Cotte's Memoirs, &c, I. 366. 



VOL. 5. 28 



