Page:Scientific Memoirs, Vol. 2 (1841).djvu/225

${\begin{aligned}&+(64\centerdot 437-79\centerdot 518e+122\centerdot 936e^{2}-152\centerdot 589e^{3})\sin \lambda \\&+(90\centerdot 184+45\centerdot 532e-185\centerdot 46e^{2})f\cos 2\lambda \\&+(14\centerdot 070-146\centerdot 386e-91\centerdot 582e^{2})f\sin 2\lambda \\&+(56\centerdot 250+0\centerdot 534e)f^{2}\cos 3\lambda \\&+(4\centerdot 188+59\centerdot 322e)f^{2}\sin 3\lambda \\&-12\centerdot 701f^{3}\cos 4\lambda \\&+16\centerdot 508f^{3}\sin 4\lambda \\Z&=-24\centerdot 593+1896\centerdot 847e+400\centerdot 343e^{2}-75\centerdot 471e^{4}\\&\qquad \qquad \qquad \qquad \qquad -544\centerdot 275e^{4}\\&+(79\centerdot 700-107\centerdot 763e+491\centerdot 744e^{2}-762\centerdot 946e^{3})f\cos \lambda \\&+(-395\centerdot 724-155\centerdot 473e+191\centerdot 176e^{2}+320\centerdot 560e^{3})f\sin \lambda \\&+(34\centerdot 187-292\centerdot 772e-228\centerdot 955e^{2})f^{2}\cos 2\lambda \\&+(-147\centerdot 439-91\centerdot 064e+212\centerdot 865e^{2})f^{2}\sin 2\lambda \\&+(5\centerdot 584+98\centerdot 870e)f^{3}\cos 3\lambda \\&+(-75\centerdot 000-0\centerdot 890e)f^{3}\sin 3\lambda \\&+20\centerdot 635f^{4}\cos 4\lambda \\&+15\centerdot 876f^{4}\sin 4\lambda {\mbox{.}}\end{aligned}}$

After these components have been calculated for a given place, we obtain in the following manner the several parts of the determination of the magnetic force, according to the customary form.

Let $\delta$ be the declination, $i$ the inclination, $\psi$ the total, and $\omega$ the horizontal intensity. Determine first $\delta$ and $\omega$ by means of the formulæ

$X=\omega \cos \delta ,\;Y=\omega \sin \delta$,

and then $i$ and $\psi$ by means of the following formulæ:

$\omega =\psi \cos i,\;Z=\psi \sin i$.

28.

As the formulæ for $X$, $Y$, $Z$, contain 71 members, their immediate calculation is a considerable labour. Its repetition for a great number of places appears the more alarming, considering that we could hardly hope to be secure from the possibility of mistake without going twice over the whole. But little would be gained by suppressing all those members of which the co-efficients are less than an integer, or even less than 10 integers, for the remaining members would still amount to 65. But as the whole value of the work would remain uncertain if not tested by a considerable number of actual observations, I have encountered the labour of calculating a table, by the assistance of which the work will be in the highest degree