VARIATION OF TERRESTRIAL MAGNETISM. 
475 
III. Expansion of Potential in tenas of a Series of Surface Ilannonics. 
We know from observation that, excepting the Arctic regions, the daily variation of 
the West force is nearly the same along' the same circle of latitude. It is this fact 
which renders the present investigation possible, as comparatively few places of 
observation will be necessary to give us a very fair idea of the nature of the oscil¬ 
lation over a considerable area of the Earth’s surface. If at any place we find that the 
daily oscillation of any one element can be expressed in the form 
cos t + sin cos '2t -f 6.3 sin -f . . ., 
when t is reckoned by local time, we may get the variation at any point of the same 
latitude circle by writing t f \ for t, where X is the longitude towards the East from 
some standard meridian and t now is the time of the standard meridian. At the time 
^ = 0 we have then the variation of the force to geographical West in different lougi- 
tudes expressed by 
Y = cos X f- sin X + a .2 cos 2X fi- h.^ sin 2X. 
The coefiicients will be functions of the latitude, and by expressing these functions 
in a series of proper form we may at once obtain an expression for the potential. 
If X is the force to geographical North, Y the force to geographical West, and Z the 
vertical force, reckoned positive upwards, we have, putting u for the colatitude, and X 
for the longitude towards the East, 
X = 
dV 
a die 
Y sin It = 
df_ 
a dx’ 
z 
dN_ 
dr’ 
a being the Earth’s radius. 
If we can expand Y sin u in terms of surface harmonics our task is accomplished, 
and for this purpose we need only express a^ sin u, sin u, &c., in a series of tesseral 
harmonics. 
The expansion of a function of an angle in terms of the trigonometrical functions of 
its multiples is so easily carried out by the method of least squares, if the function is 
given fur a regular series of submultiples of £ 77 , that it seemed to me to be the easiest 
method of proceeding to obtain first by interpolation from the observed coefiicients of 
Y its values for equidistant circles of latitude. 
To obtain a curve for each of the values a^, a^, bo, as depending on the 
latitude, we have the following data. The values are directly observed for four 
points in the Northern hemisphere. Taking the potential (as it is observed to be) 
symmetrical on the Northern and Southern hemispheres, and fixing the period to 
which we apply the calculation to be the one for which the mean value of the 
observed summer variations in the Northern hemisphere holds good, we must put, for 
the West force at the same period, in the Southern half of the globe, the observed 
ivinter values of the corresponding Northern latitudes, reversing, however, the sign to 
3 P 2 
