8 Prof. W. A. Norton on Terrestrial Magnetism. 
=Atf(Vi?+27)dx. Integrating this between the limits 0 and k, 
we have, vertical action of GB=A?t.F(l,k). Whence, vertical 
action of AB=2A¢.F(/, xk). If & may be taken sensibly the 
same for different isogeothermal lines, this expression will become 
2At.F(l). It is to be supposed, however, that the last particle 
of GB, which has a sensible action upon the needle at P, is at 
the same distance from this point whatever may be the distance 
of AB from it. The value of & will therefore be less, in propor- 
tion as the distance / is greater. Supposing the most remote par- 
ticle to be at B, and denoting its distance PB by d, k will be 
equal to Vd? —/*, and the above expression will become 2Af. F 
(1,/d? —/2), or 2At.F(2). It follows therefore that the entire 
action of any isogeothermal line AB in the vertical direction upon 
a needle at P, may be reduced to a single force, proportional to 
the temperature, and varying from one isogeothermal line to an- 
other, with the distance PG of this line from the station of the 
needle. 'The entire effect of any single lamina of matter will 
therefore be the same as if the action was confined to the parti- 
cles lying in the arc GPH; the effective force of each particle 
being proportional to its temperature, and also a certain function 
of its distance from the needle. 
This being understood, let Fig. 6. 
AB, fig. 6, represent an are " . 
crossing the parallel isogeother- eo 
mal lines at right angles, T’ 2 ? 
the mean annual temperature ° ; 
of the earth at p the station of 
the needle, ¢ and / the mean 
temperatures at the extreme 
the arc pA, and r the distance Pm. pm or y may be regarded as 
depending for its value upon Pm, Pp, and Cp; of which Pp and 
Cp are constant for the same are. Thus for any one arc, (repre- 
senting, according to what has been shown, a single lamina,) 
=9(r). If we regard the variation of temperature as uniform 
for the extent of the are AB 
iG) 
iy. ig, 9%, eh si 
“wit—T:ty:a..u=(t-T) lt ad 
a) 
a ‘ 
Whence, putting v= vertical force due to an element dy at ™% 
ymal line 
and taking the expression for the action of an isogeothe ’ 
and incorporating the 2 with the constant A, 
Thus, temperature at m=T'+(¢—T) 
