Page:Scientific Memoirs, Vol. 2 (1841).djvu/92

most probable values of those quantities. These rules are as follows.

From the quantities $R_{0}$, $R_{1}$, $R_{2}$, etc., $v_{0}$, $v_{1}$, $v_{2}$, etc., we must calculate the following expressions:

${\begin{aligned}&{\frac {\mathrm {tang} \;v_{0}}{R_{0}^{3}}}+{\frac {\mathrm {tang} \,v_{1}}{R_{1}^{3}}}+{\frac {\mathrm {tang} \,v_{2}}{R_{2}^{3}}}+{\mbox{etc.}}=A\\&{\frac {\mathrm {tang} \,v_{0}}{R_{0}^{5}}}+{\frac {\mathrm {tang} \,v_{1}}{R_{1}^{5}}}+{\frac {\mathrm {tang} \,v_{2}}{R_{2}^{5}}}+{\mbox{etc.}}=A'\\&{\frac {1}{R_{0}^{6}}}+{\frac {1}{R_{1}^{6}}}+{\frac {1}{R_{2}^{6}}}+{\mbox{etc.}}=B\\&{\frac {1}{R_{0}^{8}}}+{\frac {1}{R_{1}^{8}}}+{\frac {1}{R_{2}^{8}}}+{\mbox{etc.}}=B'\\&{\frac {1}{R_{0}^{10}}}+{\frac {1}{R_{1}^{10}}}+{\frac {1}{R_{2}^{10}}}+{\mbox{etc.}}=B'';\end{aligned}}$

thence we shall have the most probable value of

${\begin{aligned}&L={\frac {1}{2}}\cdot {\frac {AB'-A'B}{B^{\prime 2}-B^{\prime \prime 2}}}\\&{\frac {M}{T}}={\frac {1}{2}}\cdot {\frac {A'B'-AB''}{B^{\prime 2}-B^{\prime \prime 2}}}=r.\end{aligned}}$

From this, and equation (I.)

$MT={\frac {C}{t^{2}}}$

we obtain

$M={\frac {1}{t}}{\sqrt {rC}}$

(VII.)

$T={\frac {1}{t}}{\sqrt {\frac {C}{r}}}.$

(VIII.)

The experiments with the small measuring apparatus may be calculated according to these laws and formulæ, and the absolute magnetism of the bar and that of the earth determined.

 

4. Calculation, according to the above rules, of the observations made with the small measuring apparatus.

The experiments were, 1st, those of deflection, which gave the values of $u_{0}-u_{0}'$, $u_{1}-u'_{1}$, $u_{2}-u_{2}'$, $u_{2}''-u_{2}'''$, $u_{1}''-u_{1}'''$, $u_{0}''-u_{0}'''$, and the corresponding values of $R$, viz. $R_{1}$, $R_{2}$, $R_{2}$, $R_{1}$, $R_{0}$. We calculate from these the values of $v_{0}$, $v_{1}$, $v_{2}$, corresponding to $R_{0}$, $R_{1}$, $R_{2}$; and hence the values of$A$, $A'$, $B$, $B'$, $B''$, which are simple functions of the six quantities, $v_{0}$, $v_{1}$, $v_{2}$; $R_{0}$, $R_{1}$, $R_{2}$. And lastly, the value of $r$ is deduced from the quantities $A$, $A'$, $B$, $B'$, $B''$, of which it is a function. Thus, the value