Prof. W. A. Norton on Terrestrial Magnetism. 9 
g(r) 
Keys (HoT) Pe) 
=AT..F(r)do(r)+A(t— 1) Pr da(r 
r 
Integrating, v=AT f F(rjde(r)+A(t =) f Ze (r)as(r) 
Integrating between the limits Pp and PA, to obtain the force 
due to the are pA, the two integrals will become two functions 
of Pp and PA. Now, for any supposed value of Pp, PA will be 
the same at every different place on the earth, and therefore the 
values of these integrals will be every where the same. If we 
denote them by M and N, we have 
v=ATM+A(¢ —-T)N=AM.T+AN(t—T). 
By the same process we obtain for the vertical force due to the 
are pB oe =AM.T+AN(¢'—T). 
Hence the expression for the effect of the whole arc, AB, is 
v—v'=AN(t—t’)=c(t-t’) . ; ‘ 1. 
If we consider the action of a second lamina, the value of c may 
be different, but ¢—¢/ will remain very nearly the same, except 
at considerable depths where the rate of variation of the temper- 
ature may be different, or the are AB may be diminished by the 
absorption of the ethereal waves in their passage to the surface. 
If we neglect these possible variations of ¢—¢, and add together _ 
the actions of the different lamine, we obtain for the actual ver- 
tical force 
do=A(T+(¢—7) 
V=C(i—?’) ; us (2.) 
in which C is the sum of the values of ¢ for the different lamin. 
we take account of the variations of t—?’, we shall have the 
actual force equal to the sum of a series of expressions of the 
form ¢(t — ¢’ ) in which both ¢ and ¢—? will be more or less dif- 
ferent. It would seem, however, that the changes in the value — 
of t-?’, from absorption or other causes, must be very slight. 
In fact if the absorption be always a certain fractional amount of 
the intensity, there will be no change of ¢—?’ from this cause. 
It will only be necessary to regard ¢ as varying. And if the ab- 
Sorption be always the same fractional amount whatever may be 
the intensity, ¢ and therefore C will have the same value at dif- 
erent places. 
The supposition made in the investigation of formula (2), that 
the variation of temperature is uniform for the extent of he 
are AB, is not strictly true. From the equator to the latitude 
45°, and even beyond this, the rate of diminution of the tempe- 
rature for every degree of latitude continually increases. ‘The 
effect of this will be to make the vertical component somewhat 
gteater, except in the higher latitudes, than formula (2) would 
Srconp Serizs, Vol. IV, No. 10.—July, 1847. 2 
