204
C. F. GAUSS ON THE GENERAL THEORY OF
From this equation, combined with the remark in the preceding article, we obtain the general form of
. If we represent by
the following function of
,
|
![{\displaystyle {\Big (}\cos u^{n-m}-{\frac {(n-m)(n-m+1)}{2(2n-1)}}\cos u^{n-m-2}+{\frac {(n-m)(n-m-1)(n-m-2)(n-m-3)}{2\centerdot 4(2n-1)(2n-3)}}\cos u^{n-m-4}-{\&}{\mbox{c.,}}{\Big )}\sin u^{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faff77deb35a3105e1383f3c1d1d08c2aed99e96) | |
then
has the form of an aggregate of
parts,
|
![{\displaystyle P^{(n)}=g^{n,0}P^{n,0}+(g^{n,1}\cos \lambda +h^{n,1}\sin \lambda )P^{n,1}+(g^{n,2}\cos 2\lambda +h^{n,2}\sin 2\lambda )P^{n,2}+{\mbox{,}}{\&}{\mbox{c.}}+(g^{n,n}\cos n\lambda +h^{n,n}\sin n\lambda )P^{n,n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f543707abb32d74a00985156e329cfa1d87043) | |
where
,
,
,
, &c. are determinate numerical co-efficients.
If the magnetic force at the point
be resolved into three forces perpendicular to each other,
,
, and
, of which
is directed towards the centre of the earth, and
and
are tangential to a spherical surface concentric with the earth, passing through
,
directed northwards in a plane passing through
and the axis of the earth, and
directed westwards in a plane parallel to the equator of the earth, then
|
![{\displaystyle X=-{\frac {d\,V}{r\,d\,u}},\quad Y=-{\frac {d\,V}{r\sin u\,d\,\lambda }},\quad Z=-{\frac {d\,V}{d\,r}};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/098494bd12194b370b9ef7dfa61c726e577ac2be) | |
consequently,
|
![{\displaystyle {\begin{aligned}&X=-{\frac {R^{3}}{r^{3}}}\left({\frac {d\,P'}{d\,u}}+{\frac {R}{r}}\centerdot {\frac {d\,P''}{d\,u}}+{\frac {R^{2}}{r^{2}}}\centerdot {\frac {d\,P'''}{d\,u}},{\&}{\mbox{c.}}\right)\\&Y=-{\frac {R^{3}}{r^{3}\sin u}}\left({\frac {d\,P'}{d\,\lambda }}+{\frac {R}{r}}\centerdot {\frac {d\,P''}{d\,\lambda }}+{\frac {R^{2}}{r^{2}}}\centerdot {\frac {d\,P'''}{d\,\lambda }},{\&}{\mbox{c.}}\right)\\&Z={\frac {R^{3}}{r^{3}}}\left(2P'+{\frac {3RP''}{r}}+{\frac {4R^{2}P}{r^{2}}},{\&}{\mbox{c.}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2943e83113c936b821f40e8614b7bb67665ba8cb) | |
On the surface of the earth
and
are the same horizontal components which we have designated above by those letters;
is the vertical component, which is positive when directed downwards.
The expressions for these forces on the surface of the earth are, then,
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![{\displaystyle X=-\left({\frac {d\,P'}{d\,u}}+{\frac {d\,P''}{d\,u}}+{\frac {d\,P'''}{d\,u}}+{},{\&}{\mbox{c.}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3452575d6c84c077ebc15df4edccbdd072c1d857) | |