Page:Scientific Memoirs, Vol. 2 (1841).djvu/223

and we can only fill these up with the greatest uncertainty. It was soon found that the calculation must be pushed at least as far as magnitudes of the fourth order, making the number of co-efficients to be determined amount to twenty-four. In all probability, members of the fifth order will also be found influential; but, in a first attempt, the values of $k$, $m$, $k'$, &c. must be still too much charged with errors, arising from the uncertainty of many of the data (and which from their nature these values involve), to permit the introduction of a still greater number of unknown values in the process of elimination.

It should be remarked that the intensities in Sabine's map are expressed according to the arbitrary unity in common use, by which the total intensity in London $=1\centerdot 372$. In these calculations, and in the tables given in the sequel, this unity has been altered so as to make all the numbers a thousand times greater, the intensity in London on which they rest being made $=1372$. It is obvious that a unity for the intensity may be taken at pleasure, since the unity for $\mu$ may be considered as arbitrary, and made to accord therewith. If further deductions are desired requiring $\mu$ to be reduced to absolute measure, it will only be necessary to multiply all the co-efficients by the factor which reduces to an absolute measure the intensities expressed according to the arbitrary unity.

26.

The numerical values of the 24 co-efficients obtained by the first calculation, the longitude $\lambda$, being reckoned east from Greenwich, are as follows:

$g^{1\centerdot 0}=+\;925\centerdot 782\qquad \qquad$	$g^{2\centerdot 2}=+\;\;\;0\centerdot 493$
$g^{2\centerdot 0}=-\;\;\;22\centerdot 059$	$g^{3\centerdot 2}=-\;73\centerdot 193$
$g^{3\centerdot 0}=-\;\;\;18\centerdot 868$	$g^{4\centerdot 2}=-\;45\centerdot 791$
$g^{4\centerdot 0}=-\;108\centerdot 855$	$h^{2\centerdot 2}=-\;39\centerdot 010$
$g^{1\centerdot 1}=+\;\;\;89\centerdot 024$	$h^{3\centerdot 2}=-\;22\centerdot 766$
$g^{2\centerdot 1}=-\;144\centerdot 913$	$h^{4\centerdot 2}=+\;42\centerdot 573$
$g^{3\centerdot 1}=+\;122\centerdot 936$	$g^{3\centerdot 3}=+\;\;\;1\centerdot 396$
$g^{4\centerdot 1}=-\;152\centerdot 589$	$g^{4\centerdot 3}=+\;19\centerdot 774$
$h^{1\centerdot 1}=-\;178\centerdot 744$	$h^{3\centerdot 3}=-\;18\centerdot 750$
$h^{2\centerdot 1}=-\;\;\;\;\;6\centerdot 030$	$h^{4\centerdot 3}=-\;\;\;0\centerdot 178$
$h^{3\centerdot 1}=+\;\;\;47\centerdot 794$	$g^{4\centerdot 4}=+\;\;\;4\centerdot 127$
$h^{4\centerdot 1}=+\;\;\;64\centerdot 112$	$h^{4\centerdot 4}=+\;\;\;3\centerdot 175$

These numbers, which may be considered as the elements of