Page:Scientific Memoirs, Vol. 2 (1841).djvu/218

ning from one pole of the earth to the other, is sufficient for the foundation of the complete theory of the magnetism of the earth.

IV. Finally, it is clear that the complete theory is also deducible from the simple knowledge of the value of $Z$ on the whole surface of the earth. In fact, if $Z$ be developed into a series,

$Z=Q^{0}+Q'+Q''+Q'''+,$ &c.

so that the general member satisfies the often-mentioned partial differential equation; $Q^{0}$ must necessarily $=0$, and

$P'={\frac {1}{2}}Q',\quad P''={\frac {1}{3}}Q'',\quad P'''={\frac {1}{4}}Q''',$ &c.

21.

On account of the simple nature of the dependence of the several forces $X$, $Y$, $Z$, on a single function $V$, and the simple relation which they bear to each other, they are far better calculated to serve as a foundation for the theory, than the usual expression of the magnetic force by the three elements, total intensity, inclination, and declination. Or rather, the latter mode, natural as it appears in itself when the question is solely that of comprehending the facts, cannot directly serve for the foundation of the theory (at least not for the first foundation) until it has been translated into the other form.

In this view it would be very desirable that a general graphical representation of the horizontal intensity should be made; partly because it would be more immediately useful for theory than the total intensity; partly because, in far the greater number of cases, the horizontal intensity was originally that which was actually observed, the total intensity having been subsequently deduced from it by means of the dip. It is the more advisable to keep the elements of the horizontal force unmixed, as they can be determined with extreme accuracy with the present instrumental means; at any rate, the observed horizontal intensity should never be suppressed when publishing the deduced total intensity, without at least giving the dip employed in the calculation; so that a person wishing to employ the horizontal intensity for the theory may either have, or be enabled to reproduce, the original observed numbers.

Interesting as it would be to found the theory of terrestrial magnetism on observations of the horizontal needle only, and thus to anticipate the vertical part, or the inclination, it is at present