212 ON THE PERIODICITY OF THE AURORA BOREALIS. 



As the ancient astronomers, by the matchless device of cycles and epicj-cles, were 

 able to explain numerous irregularities in the apparent motions of the heavenly bodies, 

 even when referred to the earth as the centre of motion, it is evident that the general 

 formula above explained, when expanded into a sufficient number of terms, will accom- 

 plish what the theory of cycles and epicycles has accomplished, since it is only the 

 algebraical embodiment of that geometrical conception. We draw a line parallel to 

 the zero line, and at a distance from it represented by A, the length of it correspond- 

 ing to one year. On this average line slides the axis of a circle, having a radius C x and 

 turning on its axis once a year. A point in the circumference of this circle draws the 

 curve which results from the two motions, 2 n c x being the angular distance of this point 

 from the direction of the average line on January 15. On the circumference of this cir- 

 cle rests the axis of another, with a radius of C 2 , the value of 4 n c 2 expressing the posi- 

 tion for January 15 of the point on its circumference which describes the resultant curve. 

 This circle turns on its axis twice a year. On the circumference of this circle rests the 

 axis of another circle, which turns round three times in a year. 'C s and 6 n ■ c z have a 

 similar meaning to that ascribed to the corresponding constants of the other terms. 



Generally, the most conspicuous characteristics of the annual curve are indicated by 

 the values of the constants C 2 and 4 n c 2 . Compared with C x or C 3 , the value of C 2 is large 

 and the angle 4 n c 2 is such as to bring the maxima into spring and autumn and the 

 minima into summer and winter. By comparing the values of C 2 and 4 n c 2 in the curves 

 of different places, it will be seen whether there is any dependence in the times and the 

 degrees of the semi-annual fluctuations on geographical position. If, at the same place, 

 there is any change in the law of this fluctuation from century to century, it will ap- 

 pear by comparing the values of C 2 and 4 n c 2 , when computed from different series of 

 observations made at the same place in different centuries. From the magnitude of CJ, 

 especially when 2 n c x approximates to the value of 100°, the influence of longer and 

 shorter daylight in summer and winter is discovered. This influence grows to such im- 

 portance in very high latitudes that C x surpasses C 2 , and the value of 2 n c x governs the 

 times of maxima and minima. The increase of C x does not, however, progress so uni- 

 formly with increase of latitude as to warrant the inference that the second term of the 

 formula is expressive of nothing else than the influence of daylight. Superadded to 

 the semi-annual periodicity of the aurora, there is an annual periodicity exclusive of 

 that annual change introduced by the seasons. The fourth term of the formula may 

 originate in imperfect or insufficient series of observations. A periodicity of four 

 months is improbable in itself, and the co-efficient 6 re e 3 is generally small, particularly 

 in the mean curve, where the number of observations massed together is so large as to 

 mask small disturbing influences. 



