10 Prof. W. A. Norton on Terrestrial Magnetism. 
give it, (that is, supposing C to be détermined a priori. If C be 
determined from observations made at the point of maximum va- 
riation of temperature, the values of V given by equation (2) will 
be too small south of this point and too great north of it.) 
To obtain a formula for the horizontal component of the direc- 
tive force, we may proceed in the same manner as for the vertical 
component, except that we now multiply the force Pa, fig. 5, 
by the cosine of the angle aPH instead of aPC. We shall there- 
fore have for the entire action of the isogeothermal line AB, fig. 
4, the expression A’t. F(Z). Hence, that of all the isogeother- 
mal lines, or of the whole acting surface, will be reduced to that 
of the single are which crosses these lines at right angles ; the mag- 
netic intensity of the different points of this arc being propor- 
tional to the temperature, and the effective forces upon the needle 
varying according to some function of the distance. Now, as in 
the present enquiry all the active particles lie quite near to P, 
their temperatures may be considered the same and equal to that 
of the earth at the station of the needle: or, if there is a sensible 
variation at the lower layers, Fig. 7. 
the augmentation towards the : ‘ 
south will be compensated for Ses Seen — 
by an equal diminution to- 
wards the north. Hence, de- 
signating the are pm, fig. 7, 
by y, and the distance Pm by 
r, the expression for the hori- 
zontal force due to this arc is 
Jdh= fav ; BY(r )dy=A'T [ F(r do(r). 
Integrating between the limits r= Pp and r=PA, and designating 
the value of the integral by P, we have 
H’=A’T’.P; 2H’=2A’P. T 
and thus finally the total horizontal fore 
B=OT Sg eed 
This is the expression for the entire effect of a single lamina. 
For different laminz C’ may be different; and beyond a certain 
depth 'T will increase. If the supposed absorption of the mag- 
netic emanations be a certain constant fractional amount of the 
magnetic intensity of the molecules, C’ will be every where the 
same. If we take the sum of all the equations (3) answerng 
to the different lamine, we shall have an equation of the same 
form for the horizontal component of the directive force, or the 
horizontal intensity at P. It is only by comparing the results 
furnished by this equation, with observations, that we can ascer 
tain with certainty whether T' is to be taken sensibly different 
from.the mean surface temperature, and whether C’ may be te 
garded as truly constant for all places. 
