202
C. F. GAUSS ON THE GENERAL THEORY OF
The foregoing investigations apply only to the horizontal portion of the earth's magnetic force. In order to embrace the vertical force also, we must consider the problem in all its generality; therefore
must be regarded as a function of three variable magnitudes, expressing the position in space of an undetermined point
. We select for the purpose the distance
from the centre of the earth, the angle
which
makes with the northern part of the earth's axis, and the angle
, which a plane passing through
and the axis of the earth makes with a first meridian, counted as positive towards the east.
Let the function
be expanded into a series, decreasing according to the powers of
, and to which we give the following form:
|
, &c. | |
The co-efficients
,
,
, &c. are here functions of
and
; in order to see how they are connected with the distribution of the magnetic fluid in the earth, let
be an element of the earth's magnetism,
its distance from
, and let
,
,
, signify for
the same as
,
,
for
. We have then
extended so as to include every
; further
, and if
be developed in the series,
|
![{\displaystyle {\frac {1}{\rho }}={\frac {1}{r}}(T^{0}+T'{\frac {r^{0}}{r}}+T''.{\frac {r^{0}r^{0}}{r^{2}}}+{\&}{\mbox{c.}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5738597b0e4e3f6dbb9399c33ea87fb5847f38fa) | |
then
|
![{\displaystyle {\begin{aligned}&R^{2}P^{0}=-\int T^{0}\,d\,\mu ,\quad R^{3}P'=-\int T'r^{0}d\,\mu ,\\&R^{4}P''=-\int T''r^{0}r^{0}\mathrm {d} \,\mu ,\;{\&}{\mbox{c.}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3f251a8ac8577ed8ee86ef7885994d251ccfaf) | |
As
, and as according to the fundamental supposition with which we set out, the quantities of positive and of negative fluid are equal in every measureable particle in which they exist, and therefore are equal in the whole earth; that is to say,
, it follows that
|
, | |
or the first number of our series for
goes out.