Page:Scientific Memoirs, Vol. 2 (1841).djvu/245

mixed causes, we attach the same meaning as before to the signs $V$, $P^{0}$, $P'$, $P''$, &c., $v$, $p^{0}$, $p'$, $p''$, applying the former to the causes acting from within, and the latter to the causes acting from without; and if we further put $V+v=W$, $P^{0}+p^{0}=\Pi ^{0}$, $P'+p'=\Pi '$, $P''+p''=\Pi ''$, &c., then on the surface of the earth,

${\frac {W}{R}}=\Pi ^{0}+\Pi '+\Pi '',\;\mathrm {\&c.}$

where $\Pi ^{(n)}$ satisfies the same partial differential equation as $P^{(n)}$ (Art. 18.); and the two components of the horizontal magnetic force there existing are expressed by

$-{\frac {d\,W}{R\,d\,u}},\;-{\frac {d\,W}{R\sin u\,d\,\lambda }}.$

The propositions mentioned in Articles 15. and 16. retain therefore their validity in this case, and we can determine the magnitudes $\Pi '$, $\Pi ''$, $\Pi '''$, &c. simply from the knowledge of the horizontal forces, but without being able in any degree to conclude from hence only as to the existence of mixed causes. But if we consider the vertical force by itself, and bring it into the form

$Q^{0}+Q'+Q''+Q'''+,\;\mathrm {\&c.}$

so that $Q^{(n)}$ satisfies the above-mentioned partial differences, then

${\begin{aligned}&Q^{0}=P^{0}\\&Q=2P-p'\\&Q''=3P''-2p''\\&Q'''=4P'''-3p''',\;\mathrm {\&c.;} \end{aligned}}$

and, consequently,

${\begin{aligned}&3P'=\Pi '+Q',&3p'=2\Pi '-Q'\\&5P''=\Pi ''+Q'',&5p''=3\Pi ''-Q''\\&7P'''=\Pi '''+Q''',&7p'''=4\Pi '''-Q'''.\end{aligned}}$

Thus, by the combination of the horizontal force with the vertical, we obtain the means of dividing $W$ into its constituent parts $V$ and $v$, and thus of learning whether a sensible value must be assigned to the latter. Only the constant part of $v$, namely, $p^{0}$, is left wholly undetermined by the observations, the reason of which is plain from Art. 38.

Hence it appears important, in this interesting point of view likewise, to consider the horizontal magnetic force by itself, and we see in this an additional reason for the recommendations in Art. 21.