OX THE PERIODICITY OF THE AURORA BOREALIS. 209 



The values of the constants A, C z and c z are easily ohtained. Let 



< = the time of observation counted from January 1, in parts of a year as the unit, 

 = the number of observed auroras corresponding to this time ; 



if -S' = represents the sum of all the values of N which correspond to the different 

 values of /, the following equations will be found : 



2. m A = S X, , in which m represents the number of interval.--, that is, the number of months in the year. 



3. m C x cos 2 7t xc x = 2 S N, sin. 2 n x t. 



4. m C x sin. 2 n x c x = 2 S Nt cos. 2 n x t. 



For, 



5. N = A -f S C x sin. 2 n x (t + c x ). 



= ^ + 5 ( C x sin. 2 n x t cos. 2 n x c x -f- cos. 2 n x t sin. 2nx c x ). Hence, 

 S X t = m A, 



because the sum of the values of sin. 2 ti x t and cos. 2 ti x t, on account of their differ- 

 ent signs plus or minus, equals zero. 



Multiplying each side of equation (5) by sin. 2 ti x t, 



6. Xsin. 2 n x t— A sin. 2 n x t -4- 5 C x sin. (2 7r x <) 2 cos. 2;tx c^-|-,S C x sin. 2 » a; < cos. 2 n ar < sin. 2 « x c„ 



The average value of the first and last terms on the right is zero, since the average- 

 value of the sine itself is zero. The second term is always plus, because it is the 

 square of the sine. 



Sin. 2 n x t sin. 2 n x t = — \ [cos. (2 n x I -\-2 n x t) — cos. (2 n x I — 2 n x <)]. = — \ cos. \n x t-\-\. 



The first term — £ cos. in xt averages zero for all the different values of (J). There 

 remains the factor i for each value of t. Hence, 



3. S N t • sin. 2 n x t = „ C x cos. 2 n x c*. 

 Multiplying each side of equation (5) by cos. 2 ti x t and reducing as before, 



4. S Nt cos. 2 7t x t= ^ C' x sin. 2 n x c^ 



The values of c z are found by dividing equation (4) by equation (3). 



S X: COB. 2 71 X t 



7. tang. 2 ti x c x = —— . 



.S N t sin. 2 7i x t 



VOL. X. 27 



