Page:Scientific Memoirs, Vol. 2 (1841).djvu/89

The ratio which the force of the bar on the needle (at the given distance $R$) bears to that of the earth's magnetism is expressed by the quotient

${\frac {F}{mT}}$

and according to equation (III.)

$mt={\frac {F}{\mathrm {tang} \,v}},\quad {\mbox{or}}\quad {\frac {F}{mT}}=\mathrm {tang} \,v.$

But according to equation (II.)

$mM{\frac {FR^{3}}{2}},\quad {\mbox{or}}\quad {\frac {F}{mT}}={\frac {2M}{R^{3}T}},$

and,

${\frac {2M}{R^{3}T}}.MT={\frac {C}{t^{2}}}\cdot \mathrm {tang} \,v,$

makes known to us the maximum moment of rotation which the bar with the magnetism $M$ would exert on a similar bar at the distance $R$; for this maximum, according to the fundamental laws of magnetism, must be ${\frac {2M^{2}}{R^{3}}}$; and the above equation, gives ${\frac {2M^{2}}{R^{3}}}={\frac {C}{t^{2}}}.\mathrm {tang} \,v$.

This result, multiplied by the cube of the distance $R$, gives double the moment of rotation; and the square root of the half gives the force of the bar in absolute measure; or

$M={\frac {1}{t}}{\sqrt {\frac {CR^{3}.\mathrm {tang} \,v}{2}}}.$

(V.)

If, finally, we divide by the moment of rotation exerted by the earth on the needle, calculated according to equation (I.) we obtain

$T={\frac {1}{t}}{\sqrt {\frac {2C}{R^{3}\mathrm {tang} \,v}}},$

(VI.)

and this number expresses the absolute measure of the earth^s magnetism.

 

"This appears the most easily understood exposition which can be given, without the use of mathematical signs, of the possibility of expressing the force of the earth's magnetism by a number which shall be perfectly independent of the individuality of the magnetic bars employed. In the actual application some points will appear in a somewhat different form, without, however, affecting in the least the nature of the method; and it will, besides, be necessary to take into consideration several collateral circumstances."