ON GAUSS’S THEORY AND TERRESTRIAL MAGNETISM. 171 
among these unknown coefficients; and for each place at which the three 
elements are known we have three such equations. Hence to obtain the 
general expressions of X, Y, Z, to the fourth order inclusive, it is theoretically 
sufficient to know the three elements at eight points on the earth’s surface. 
But, owing to the errors of observation, and to the influence of the terms 
neglected in the approximation, the number of determinations must, in prac- 
tice, be much greater than the number of unknown coefficients. 
The foregoing conclusions are based upon the hypotheses that magnetic 
attraction and repulsion vary according to the inverse square of the distance, 
and that the magnetic action of the globe is the resultant of the actions of all 
its parts. It is likewise assumed that there are two magnetic fluids in every 
magnetizable element, and that magnetization consists in their separation. 
But for these hypotheses we may substitute that of Ampére, which supposes 
the magnetic force to be due to electric currents circulating round the mole- 
cules of bodies. 
This theory may be applied to the changes of terrestrial magnetism, whe- 
ther regular or irregular, provided only that the causes of these changes act 
in the same manner as galvanic currents, or as separated magnetic fluids. 
We have only to consider whether the data which we possess are sufficient 
for such an application. 
It has been already stated that, for the general determination of X, Y, and 
Z, we must know their values at eight points (at least) on the earth’s sur- 
face, these points being as widely distributed as possible. The same thing 
holds with respect to the changes 6X, dY, 6Z; and to apply the formule so 
determined, and to compare them with observation, corresponding values 
must be known for (at least) one more point. In the case of the irregular 
changes these observations must, of course, be simultaneous. The regular 
changes must be inferred from observations extending over considerable 
periods ; and there is reason to believe that these periods must be identical, 
or nearly so, for all the stations, since the changes are known to vary from 
month to month and from year to year. 
The regular variations of the three elements X, Y, Z, or their theoretical 
equivalents, have been obtained by observation, for nearly the same period, 
at Greenwich, Dublin, and Makerstoun, in the British Islands; at Brussels 
and Munich, on the Continent of Europe; at Toronto and Philadelphia, in 
North America ; at Simla, Madras, and Singapore, in India ; and at St. Helena, 
the Cape of Good Hope, and Hobarton, in the southern hemisphere. Of these 
thirteen stations, however, the three British must be regarded, for the pre- 
sent purpose, as equivalent to one only, on account of their proximity; and 
the same thing may be said of the two North American stations and of the 
two stations in Hindostan. This reduces the number of available stations to 
nine, the minimum number required for the theoretical solution of the pro- 
blem in the degree of approximation already referred to, and considered by 
Gauss to be necessary. It is true that we may add to these the stations at 
which two only of the three elements have been observed, viz. Prague and 
St. Petersburg, the three Russian stations in Siberia, and Bombay. But even 
with this addition, the number is probably insufficient for the satisfactory 
determination of the unknown coefficients; for it is to be remembered that 
the places, few as they are, are not distributed with any approach to uni- 
formity, and that very large portions of the globe are wholly unrepresented 
by observations. 
For the reason already stated, this defect in the existing data cannot be 
now repaired by supplemental observations at new stations, unless the series 
