Page:Scientific Memoirs, Vol. 2 (1841).djvu/91

"One main difficulty in the application of this method consists in the fact, that the above-mentioned law holds good, namely, that the action of a magnetic needle is inversely as the cube of the distance, with sufficient accuracy only for very great distances, and where the effects are too small to be determined with precision by direct observation. At moderate distances the variations from the law become very perceptible; but theory teaches that these very differences are subject to rule; and mathematics afford us the means of recognising, and almost wholly eliminating them, by the combination of experiments made at various moderate distances."

 

For the purpose of showing the application of the small measuring apparatus to the above-mentioned observations, we shall give lastly, in a few words, the necessary process of correction. This is threefold:

1. Instead of the values given by direct observation for the deflexions $v_{0}$, $v_{1}$, $v_{2}$, etc., of the needle by the magnet bar acting at various distances, $R_{0}$, $R_{1}$, $R_{2}$, etc., the following combined values are to be taken:

${\begin{aligned}&v_{0}={\frac {1}{4}}(u_{0}^{}-u'_{0}+u''_{0}-u'''_{0})\\&v_{1}={\frac {1}{4}}(u_{1}-u'_{1}+u''_{1}-u'''_{1})\\&v_{2}={\frac {1}{4}}(u_{2}^{}-u'_{2}+u''_{2}-u'''_{2}){\mbox{ etc.}}\end{aligned}}$

2. To the approximate values of${\frac {M}{T}}$, which were obtained by equation (IV.)

${\frac {M}{T}}={\frac {R^{3}\,\mathrm {tang} \,v}{2}}$

the following corrections are added:

${\begin{aligned}&{\mbox{ Approximate value of}}{\frac {M}{T}}.&Correction.\\&{\frac {R_{0}^{3}\,\mathrm {tang} \,v_{o}}{2}}&-{\frac {L}{R_{0}^{2}}}\\&{\frac {R_{1}^{3}\,\mathrm {tang} \,v_{1}}{2}}&-{\frac {L}{R_{1}^{2}}}\\&{\frac {R_{2}^{3}\,\mathrm {tang} \,v_{2}}{2}}{\mbox{etc.}}&-{\frac {L}{R_{2}^{2}}}{\mbox{etc.}}\end{aligned}}$

3. s the number of the measured dimensions $R_{0}$, $R_{1}$, $R_{2}$, etc. and $v_{0}$, $v_{1}$, $v_{2}$, etc., is greater than is required for the determination of the unknown quantities $L$ and ${\frac {M}{T}}$, the rules of the calculus of probabilities are employed in order to obtain from them the