ON THE EARTH’S MAGNETISM. 
383 
With these coefficients (and neglecting the non-periodic part of the phenomena) have 
been calculated the ordinates for the construction of the thick curves (Plate 53. figs. 1-12), 
the ordinates of which represent disturbance, and the abscissae time. 
19. It may be objected to the procedure thus far, that the application of Bessel’s. 
method to any arbitrary series of periodical numbers would yield a smooth-flowing 
curve, although the numbers themselves were subject to no corresponding law: this, we- 
reply, is a mistake; the law is inherent in the series of numbers. It is another 
question to what cause the law must, in a particular case, be attributed ; but this is so 
also when a periodical law has been found in a series of observations, by applying the 
common method of finding average values at different phases of the period. It may be 
interesting to some of our readers to show that, where the circumstances allow of the 
application of the latter method, it leads to the same form of curve as Bessel’s process. 
We choose for this purpose the variations, with the sidereal period of Mercury, of 
disturbances of Declination (Easterly and Westerly) and of disturbances increasing and 
decreasing the Horizontal Force. If we take twenty-six equidistant times in the period 
of Mercury and twenty-six consecutive months, the several months will correspond to 
the twenty-six phases of Mercury’s period, as shown below. 
Twenty-sixths of the period of Mercury . . . 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
Months 
0 
3 
6 
9 
12 
15 
18 
21 
24 
1 
4 
7 
10 
Twenty-sixths of the period of Mercury . . . 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
J 
Months 
13 
16 
19 
22 
25 
2 
5 
8 
11 
14 
17 
20 
23 
And arranging each successive twenty-six months’ aggregates in this way and in 
successive lines, we get, for each phase, eleven observed disturbance-aggregates, of which 
averages are calculated. Means are then taken of each consecutive pair of these 
averages, forming twenty-six new averages, and this process is repeated six times ; after 
this the means are taken of every consecutive three of the last averages, and these 
numbers are curved thin in figs. 1-4. It will be seen that they agree with the thick 
curves obtained, by Bessel’s process, which are also constructed from twenty-six equi- 
distant ordinates ; but the agreement is closer, as it clearly should be, when the twenty- 
six calculated ordinates are treated in the same manner (described above) as the twenty- 
six average disturbance-aggregates were, to obtain the ordinates of the thin curves. In 
this way the ordinates of the dotted curves have been obtained ; and although the thick 
curves must be taken as best representing the true law, the dotted ones are more directly 
comparable with the thin curves, having been obtained by a similar process. The slight 
disagreement that is observable must be attributed mainly to the omission of the fourth 
and higher pairs of terms of Bessel’s expression. 
