Page:Scientific Memoirs, Vol. 2 (1841).djvu/244

the former supposition, $v$ must first be considered as a function of both $r$, $u$, and $\lambda$; it must be differentiated according to $r$, and then $r=R$ must be substituted.

But for the inner space $S'$, to which the surface of the earth belongs, $v$ can only be developed in a series according to ascending powers of $r$. If we make

${\frac {V}{R}}=p^{0}+{\frac {r}{R}}\centerdot p'+{\frac {r^{2}}{R^{2}}}\centerdot p''+{\frac {r^{3}}{R^{3}}}\centerdot p''',\;\mathrm {\&c.}$

$p^{0}$ is a constant magnitude, namely, the value of ${\frac {V}{R}}$ at the centre of the earth; $p'$, $p''$, $p'''$, &c., on the other hand, are functions of $u$, and $\lambda$, which satisfy the same partial differential equations as $p'$, $p''$, $p'''$, above.

Hence it follows, in a similar manner to Art. 20, that the knowledge of the value of $v$ at every point of the earth's surface is sufficient to enable us to deduce therefrom the general expresssion for the space $S'$; that we may arrive at the knowledge of this value with the exception of a constant part,—or, which is the same thing, at the knowledge of the co-efficients $p'$, $p''$, $p'''$, &c.,—by the knowledge of the horizontal forces on the surface of the earth; but that the value of the vertical force on the surface of the earth is not

$=2p'+3p''+4'''+{},\;\mathrm {\&c.}$

as it would be if the forces acted outwards from the interior of the earth, but is

$=-p'-2p''-3p'''-{},\;\mathrm {\&c.}$

Now, as our numerical elements (Art. 26.), determined under the supposition of the first formula, give a very satisfactory representation of the phenomena generally, whereas, the phenomena would be wholly incompatible with the second formula, the fallacy of the hypothesis, which places the causes of terrestrial magnetism in space external to the earth, may be looked upon as proved.

40.

At the same time, this must not be looked upon as decidedly disproving the possibility of a part, though comparatively a very small part, of the terrestrial magnetic force proceeding from the upper regions: a far more full and far more accurate knowledge of the phenomena may in future throw light on this important point of theory. If, under the supposition of