Page:Scientific Memoirs, Vol. 2 (1841).djvu/93

of $r$ is obtained by calculation from the experiments of deflection.

2nd. From the experiments of vibration the value of the time of vibration $t$ is found: having thus the values of $r$ and $t$, it is only required, for the purposes of the travelling observer, to calculate

${\frac {1}{t{\sqrt {r}}}};$

for this value is proportional to the number, which expresses the absolute terrestrial magnetism, and consequently suffices for the comparison of the absolute intensity of all places where such experiments may be performed. Such a comparison is usually the only object sought by the travelling observer. It may sometimes, however, be desirable to obtain not only comparisons of the absolute intensity at various places, but the absolute intensity itself; the apparatus may be lost on a voyage, and be replaced by a new one; and it then becomes necessary, in order to compare the two series of results obtained with instruments which cannot be compared together, to calculate the moment of inertia of the magnet bar, the time of vibration of which had been observed, and to extract its square root. The product of the quantity ${\frac {1}{t{\sqrt {r}}}}$ into the square root, and into the number $\pi ={}$ 3·14159 … gives a number expressing the earth's magnetism in absolute measure.

On this account it is advantageous that the bar should be an accurate parallelopiped, because in such case the moment of inertia can be deduced for the present purpose directly from the weight $p$, the length $a$, and the breadth $b$ of the bar. For it is well known that the square $a^{2}+b^{2}$ of the diagonal of the superficies of the parallelopiped, multiplied by the mass $p$ of the weight, and divided by 12, gives the moment of inertia sought, in the case in which the bar shall have been suspended by the centre of that superficies. Consequently in the equations (VII.) and (VIII.)

$C=9\cdot 8696\ldots {\frac {a^{2}+b^{2}}{12}}\cdot p$

If we compare the observations above mentioned with these formulæ, it will be seen that the following quantities have been directly measured, and the following values found for them: