210 ON THE PERIODICITY OF THE AURORA BOREALIS. 



2V cos. + 2^ cos. x 30° + N 2 cos. x GO + 2V 3 cos. x 90° -f ZV 4 cos. x 120° &c. 

 = y o sin. 4- •# i sin. x 30° + 2V, cos. x 60° + N t cos. a; 90° + & t cos. a; 120° &c/ 



Multiplying equation (3) by cos. 2 it x c x and equation (4) by sin. 1 n x c z and then 

 adding them, 



8. m C x = 2 cos. 2 7T x c x S 2i, sin. 2 tt a; t -4- 2 sin. 2 n x c x S N, cos. 2 n x t, 



from which the values of C x may be obtained. 



If the series is supposed to converge so rapidly that the first four terms will give the 

 number of auroras to a sufficiently close approximation, comparable with the accuracy 

 of observation, the general formula becomes 



9. N= A + d sin. 2 n (t -\- <^) + O a sin. 4s(( + f2 )+ C 8 sin. 6 a (t + c 3 ). 



By this general formula I have calculated the mean annual curve (in this case alone 

 carrying the computation to the seventh term), and also the particular curve for those 

 places where a sufficient number of auroras had been observed to encourage such an 

 undertaking. The number of auroras for each month has then been computed by the 

 formula and that number compared with the number actually observed, and from the 

 square of the differences the mean probable error has been obtained by the following 

 formula : 



; SA 2 

 10. e: 



I SA 2 

 I = ,674 y m — 1. 



If we differentiate equation (9), and make the differential equal to zero, we shall have 

 an equation from which the days of maxima and minima of auroral display during the 

 year may be computed. 



11. dcos. 2 n{t-\- Cl ) +2 (7 2 cos. 4 n (t + c„) -\- 3 C 3 cos 6 n (t-\-c s ) = 0. 



The direct calculation of (t) from this equation would be difficult. Knowing from 

 the observations themselves nearly at what times these maxima and minima occur, by 

 substituting empirical values for (t) we arrive by this tentative process to the required 

 result : viz., the precise times and the values of the maxima and minima. In all the pre- 

 ceding computations, whatever the number of auroras actually observed in the different 

 months, they have been reduced to what they would have been if exactly one hundred 

 auroras had been seen during the year, so that in each curve the value of A is 8.33+. 



