Page:Scientific Memoirs, Vol. 2 (1841).djvu/224

the theory of terrestrial magnetism, are used both here and in the formation of the table to be described in the sequel, just as they were given by calculation, without omitting decimals. To any one conversant with calculation it is superfluous to remark, that these fractional parts have in themselves no value, as we are still far from being able to eliminate with certainty even the integers. But it is important that the observations should be closely compared with one and the same definite system of elements; and, as by leaving out decimals nothing would be gained in point of convenience in computing, there was no reason for altering in any respect the elements given by calculation.

27.

The expression for $V$, developed according to the above numbers, is as follows: for the sake of brevity $e$ stands for $\cos u$, and $f$ for $\sin u$.

${\begin{aligned}{\frac {V}{R}}&=-1\centerdot 977+937\centerdot 103e+71\centerdot 245e^{2}-18\centerdot 868e^{3}\\&\qquad \qquad \qquad \qquad \qquad \qquad -108\centerdot 855e^{4}\\&+(64\centerdot 437-79\centerdot 518e+122\centerdot 936e^{2}+152\centerdot 589e^{3})f\cos \lambda \\&+(-188\centerdot 303-33\centerdot 507e+47\centerdot 794e^{2}+64\centerdot 112e^{3})f\sin \lambda \\&+(7\centerdot 035-73\centerdot 193e-45\centerdot 791e^{2})f^{2}\cos 2\lambda \\&+(-45\centerdot 092-22\centerdot 766e-42\centerdot 573e^{2})f^{2}\sin 2\lambda \\&+(1\centerdot 396+19\centerdot 774e)f^{3}\cos 3\lambda \\&+(-18\centerdot 750-0\centerdot 178e)f^{3}\sin 3\lambda \\&+4\centerdot 127f^{4}\cos 4\lambda \\&+3\centerdot 175f^{4}\sin 4\lambda \mathrm {.} \end{aligned}}$

We may here add the completely developed expressions for the three components of the magnetic force.

${\begin{aligned}X&=(937\centerdot 103+142\centerdot 490e-56\centerdot 603e^{2}-435\centerdot 420e^{3})f\\&+(-79\centerdot 518+181\centerdot 435e-298\centerdot 732e^{2}-368\centerdot 808e^{3}\\&\qquad \qquad \qquad \qquad \qquad \qquad +610\centerdot 357e^{4})\cos \lambda \\&+(-33\centerdot 507+283\centerdot 892e+259\centerdot 349e^{2}-143\centerdot 383e^{3}\\&\qquad \qquad \qquad \qquad \qquad \qquad -256\centerdot 448e^{4})\sin \lambda \\&+(-73\centerdot 193-105\centerdot 652e+219\centerdot 579e^{2}+183\centerdot 164e^{3})f\cos 2\lambda \\&+(-22\centerdot 766+175\centerdot 330e+68\centerdot 098e^{2}-170\centerdot 292e^{3})f\sin 2\lambda \\&+(19\centerdot 774-4\centerdot 188e-79\centerdot 096e^{2})f^{2}\cos 3\lambda \\&+(-0\centerdot 178+56\centerdot 250e+0\centerdot 716e^{2})f^{2}\sin 3\lambda \\&-16\centerdot 508ef^{3}\cos 4\lambda \\&-12\centerdot 701ef^{3}\sin 4\lambda \\Y&=(188\centerdot 303+33\centerdot 507e-47\centerdot 794e^{2}-64\centerdot 112e^{3})\cos \lambda \end{aligned}}$