Page:Scientific Memoirs, Vol. 2 (1841).djvu/221

of the degree of uncertainty which still attaches to the fundamental members.

From the equations

${\begin{aligned}&k=-g^{1\centerdot 0}{\frac {d\,P^{1\centerdot 0}}{d\,u}}-g^{2\centerdot 0}{\frac {d\,P^{2\centerdot 0}}{d\,u}}-g^{3\centerdot 0}{\frac {d\,P^{3\centerdot 0}}{d\,u}}-,\;\mathrm {\&c.} \\&m=2g^{1\centerdot 0}P^{1\centerdot 0}+3g^{2\centerdot 0}P^{2\centerdot 0}+4g^{3\centerdot 0}\rho ^{3\centerdot 0}+,\;\mathrm {\&c.} {\mbox{,}}\end{aligned}}$

the total number of which is double the number of the parallels, we have to obtain, by the method of least squares, (after substituting in ${\frac {d\,P^{1\centerdot 0}}{d\,u}}$, &c., and in $P^{1\centerdot 0}$, &c. the corresponding numerical values of $u$,) as many of the co-efficients $g^{1\centerdot 0}$, $g^{2\centerdot 0}$, $g^{3\centerdot 0}$, &c. as require to be taken into account.

In like manner the equations

${\begin{aligned}-&k'=g^{1\centerdot 1}{\frac {d\,P^{1\centerdot 1}}{d\,u}}+g^{2\centerdot 1}{\frac {d\,P^{2\centerdot 1}}{d\,u}}+g^{3\centerdot 1}{\frac {d\,P^{3\centerdot 1}}{d\,u}}+,\;\mathrm {\&c.} \\&L'=g^{1\centerdot 1}{\frac {P^{1\centerdot 1}}{\sin u}}+g^{2\centerdot 1}{\frac {P^{2\centerdot 1}}{\sin u}}+g^{3\centerdot 1}{\frac {P^{3\centerdot 1}}{\sin u}}+,\;\mathrm {\&c.} \\&m'=2g^{1\centerdot 1}P^{1\centerdot 1}+3g^{2\centerdot 1}P^{2\centerdot 1}+4g^{3\centerdot 1}P^{3\centerdot 1}+,\;\mathrm {\&c.} {\mbox{,}}\end{aligned}}$

the number of which is three times as great as the number of parallels, serve to determine the co-efficients $g^{1\centerdot 1}$, $g^{2\centerdot 1}$, $g^{3\centerdot 1}$, &c. And the following,

${\begin{aligned}-&K'=h^{1\centerdot 1}{\frac {d\,P^{1\centerdot 1}}{d\,u}}+h^{2\centerdot 1}{\frac {d\,P^{2\centerdot 1}}{d\,u}}+h^{3\centerdot 1}{\frac {d\,P^{3\centerdot 1}}{d\,u}}+,\;\mathrm {\&c.} \\-&l'=h^{1\centerdot 1}{\frac {P^{1\centerdot 1}}{\sin u}}+h^{2\centerdot 1}{\frac {P^{2\centerdot 1}}{\sin u}}+h^{3\centerdot 1}{\frac {P^{3\centerdot 1}}{\sin u}}+,\;\mathrm {\&c.} \\&M'=2g^{1\centerdot 1}P^{1\centerdot 1}+3h^{2\centerdot 1}P^{2\centerdot 1}+4h^{3\centerdot 1}P^{3\centerdot 1}+,\;\mathrm {\&c.} \end{aligned}}$

determine the coefficients $h^{1\centerdot 1}$, $h^{2\centerdot 1}$, $h^{3\centerdot 1}$, &c. Further, the equations

${\begin{aligned}-&k''=g^{2\centerdot 2}{\frac {d\,P^{2\centerdot 2}}{d\,u}}+g^{3\centerdot 2}{\frac {d\,P^{3\centerdot 2}}{d\,u}}+g^{4\centerdot 2}{\frac {d\,P^{4\centerdot 2}}{d\,u}}+,\;\mathrm {\&c.} \\&L''=2g^{2\centerdot 2}{\frac {P^{2\centerdot 2}}{\sin u}}+2g^{3\centerdot 2}{\frac {P^{3\centerdot 2}}{\sin u}}+2g^{4\centerdot 2}{\frac {P^{4\centerdot 2}}{\sin u}}+,\;\mathrm {\&c.} \\&m''=3g^{2\centerdot 2}P^{2\centerdot 2}+4g^{3\centerdot 2}P^{3\centerdot 2}+5g^{4\centerdot 2}P^{4\centerdot 2}+,\;\mathrm {\&c.} \end{aligned}}$

determine the co-efficients $g^{2\centerdot 2}$, $g^{3\centerdot 2}$, $g^{4\centerdot 2}$, &c.; and we obtain the co-efficients of the succeeding higher numbers in a similar manner.

24.

The chief advantage which this method possesses over that