Page:Scientific Memoirs, Vol. 2 (1841).djvu/206

It is obvious that in this equation the units of intensity and of distance are arbitrary.

10.

As an example, we will apply the formula to the magnetic elements of

Göttingen	$\delta ^{0}=18^{rc}\;38'$	$i^{0}=67^{\circ }\;56'$	$\psi ^{0}=1.357$
Milan	$\delta '=18\;\;33$	$i'=63\;\,49$	$\psi '=1.294$
Paris	$\delta ''=22\;\;04$	$i''=67\;\;24$	$\psi ''=1.348$

whence it follows that

${\begin{aligned}&\omega ^{0}=0.50980\\&\omega '=0.57094\\&\omega ''=0.51804\end{aligned}}.$

Taking the geographical position of

Göttingen	51°	32′	latitude	9°	58′	longitude from Greenwich
Milan	45	28		9	09
Paris	48	52		2	21

and performing the calculation for a spherical surface only, we find

(01)	=	5°	11′	31″	$P^{0}P'$	=	6°	5′	20″
(10)	=	184	35	35
(12)	=	128	47	31	$P'P''$	=	5	41	06
(21)	=	303	48	01
(20)	=	238	20	20	$P^{0}P''$	=	5	32	04
(02)	=	64	10	12

Substituting these values in our equation, and those given above for $\delta ^{0}$, $\delta '$, $\delta ''$, we have

${\begin{aligned}&0=17556\;\omega ^{0}+2774\;\omega '-20377\;\omega '',\\&{\text{or, }}\omega ''=0.86158\;\omega ^{0}+0.13613\;\omega '.\end{aligned}}$

Hence we deduce from the observed horizontal intensities at Göttingen and Milan, that at Paris $\omega ''=0.51696$, agreeing almost exactly with the observed value $0.51804$.

It is easily seen that if we permit ourselves to take the distances $P^{0}$, $P''$, &c. instead of their sines, the above formula can be expressed immediately by the geographical longitudes and latitudes of the places.

11.

The line on the earth's surface, in all points of which $V$ has the same value $=V^{0}$, divides generally speaking the parts of the surface in which the value of $V$ is greater than $V^{0}$, from those in