Page:Scientific Memoirs, Vol. 2 (1841).djvu/226

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214

C. F. GAUSS ON THE GENERAL THEORY OF

abridged and facilitated, and at the same time the important object of security against errors of calculation will be materially promoted.

For the construction of the table the values of the coefficients were brought into the following form:

{\begin{aligned}&X=a^{0}+a'\cos(\lambda +A')+a''\cos(2\lambda +A'')+\\&\qquad \qquad a'''\cos(3\lambda +A''')+a^{\mathrm {IV} }\cos(4\lambda +A^{\mathrm {IV} })\\&Y=b'\cos(\lambda +B')+b''\cos(2\lambda +B'')+\\&\qquad \qquad b'''\cos(3\lambda +B''')+b^{\mathrm {IV} }\cos(4\lambda +B^{\mathrm {IV} })\\&Z=c^{0}+c'\cos(\lambda +C')+c''\cos(2\lambda +C'')+\\&\qquad \qquad c'''\cos(3\lambda +C''')+c^{\mathrm {IV} }\cos(4\lambda +C^{\mathrm {IV} }).\end{aligned}}

The first table contains those parts of $X$ and $Z$ which are independent of $X$ . In the four next tables are found the values of the auxiliary angles $A'$ , $A''$ , &c., and the logarithms of $a'$ , $a''$ , &c., all for the several degrees of latitude $\phi =90^{\circ }-u$ . The table is placed at the end of the memoir.

The calculation for Göttingen is given as an example.

For latitude $51^{\circ }\;32'$ we find from the tables:

$a^{0}=+500.8$		$c^{0}=+1465\centerdot 2$
$\log a'=2.28980$	$\log b'=2.18900$	$\log c'=2.20204$
$\log a''=1.79403$	$\log b''=2.03220$	$\log c''=2.12777$
$\log a'''=1.32522$	$\log b'''=1.46845$	$\log c'''=1.43199$
$\log a^{\mathrm {IV} }=0.59391$	$\log b^{\mathrm {IV} }=0.70016$	$\log c^{\mathrm {IV} }=0.59091$
$A'=249^{\circ }30'$	$B'=358^{\circ }24'$	$C'=105^{\circ }44'$
$A''=311\;\;45$	$B''=64\;\;50$	$C''=165\;\;15$
$A'''=234\;\;10$	$B'''=318\;\;13$	$C''=42\;\;22$
$A^{\mathrm {IV} }=142\;\;26$	$B^{\mathrm {IV} }=232\;\;26$	$C^{\mathrm {IV} }=322\;\;26$