Page:Scientific Memoirs, Vol. 2 (1841).djvu/220

might have a very injurious effect on the results of the elimination^[1].

To diminish the unfavourable effect of these circumstances, the number of series of observations from stations well distributed over the whole globe ought to be much greater than that of the unknown values, and these should be derived from the observations by the method of least squares. As all the equations are only linear, the process would, it is true, be uniform; but the extent of the labour, arising from the great number of unknown values and equations, would be such as might well deter the most courageous calculator from undertaking it in this form, especially as the result might be wholly vitiated by the introduction either of defective observations or of accidental errors of calculation.

23.

There is another mode of proceeding, which, as it is free from a part of these difficulties, appears better adapted for a first trial. We shall develope it in this place without omitting to notice objections to which its application may be liable in the present state of the inquiry. This method supposes the knowledge of all three elements at points so grouped on a sufficient number of parallels as to divide them into a sufficient number of equal portions. The numerical values of $X$, $Y$, and $Z$, are to be first deduced from the given elements of the usual form.

The values of $X$, $Y$, $Z$, are then brought by the known method in each parallel to the form

${\begin{aligned}&X=k+&k'\cos \lambda +K'\sin \lambda +k''\cos 2\lambda +K''\sin 2\lambda \\&&+k'''\cos 3\lambda +K'''\sin 3\lambda +\mathrm {\&c.} \\&Y=l+&l'\cos \lambda +L'\sin \lambda +l''\cos 2\lambda +L''\sin 2\lambda \\&&+l'''\cos 3\lambda +L'''\sin 3\lambda +,\,\mathrm {\&c.} \\&Z=m+&m'\cos \lambda +M'\sin \lambda +m''\cos 2\lambda +M''\sin 2\lambda \\&&+m'''\cos 3\lambda +M'''\sin 3\lambda +\mathrm {,\&c.} \end{aligned}}$

We then obtain as many values for each of the co-efficients $k$, $l$, $m$, $k'$, &c., as there are parallels of latitude under consideration.

Theory would give in each parallel $l=0$; therefore the values of $l$ which result from the calculation furnish a kind of measure

1. In such a mode of determination, the effect of these circumstances would be least injurious if the eight points were distributed symmetrically on the surface of the earth; that is to say, if they coincided, or nearly so, with the corners of a cube inscribed in the globe.