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XV. The Diurnal Variation of Terrestrial Magnetism. 
By Arthur Schuster, F.R.S., Professor of Physics in Oivens College. With an 
Appendix hy H. Lamb, F.R.S., Professor of Mathematics in Owens College. 
Received Marcli 20,—Read March. 28, 1889. 
I. Introduction. 
In the year 1839 Gauss published his celebrated Memoir on Terrestrial Magnetism, 
in which the potential on the Earth’s surface was calculated to 26 terms of a series of 
surface harmonics. It was proved in this Memoir th.at, if the horizontal components 
of magnetic force were known all over the Earth, the surface potential could be derived 
without the help of the vertical forces, and it is well known now how these latter can 
be used to separate the terms of the potential which depend on internal from those 
which depend on external sources. Nevertheless Gauss made use of the vertical 
forces in his calculations of the surface potential in order to ensure a greater degree 
of accuracy. He assumed for this purpose that magnetic matter was distributed 
through the interior of the Earth, and mentions the fair agreement between calculated 
and observed facts as a justification of his assumption. In the latter part of the 
Memoir it was suggested that the same method should be employed in the investiga¬ 
tion of the regular and secular variations. 
The use of harmonic analysis to separate internal from external causes has never 
been put to a practical test, but it seems to me to be especially well adapted to 
enquiries on the causes of the periodic oscillations of the magnetic needle. 
If the magnetic effects can be fairly represented by a single term in the series of 
harmonics as far as the horizontal forces are concerned, there should be no doubt as to 
the location of the disturbing cause, for the vertical force should be in the opposite 
direction if the origin is outside from what it should be if the origin is inside the 
Earth. As the expression for the potential contains in one case the distance from the 
Earth’s centre in the numerator, in the other case in the denominator, and as the vertical 
force depends on the differential coefficient with regard to distance from the Earth’s 
centre, each single term in the series is of opposite sign according to the location of the 
cause ; but what is true for each single term need not be true for the sum of the series. 
By a curious combination of terms the vertical forces might possibly be of the same 
sign, on whichever of the two hypotheses it is calculated. In any case, however, the 
differences between the two results will be of the same order of magnitude as the 
vertical force itself. If it is then a question simply of deciding whether the cause is 
outside or inside, without taking into account a possible combination of both causes, 
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