same 



Prof. W. A. Norton on Terrestrial Magnetism. 9 



rft;=A^T+(<-T)^ ) )p'(r>^=A(T+(<-T)^- ) )F'(r)dT(r) 



AT . F'(r)d?{r)+A(*-T)— T'(r>fa(r). 



Cv 



Integrating, v=ATJ F'(r)tf v (r)+A(* -T)J 



Integrating between the limits Pp and PA, to obtain the force 

 due to the arc pA, the two integrals will become two functions 

 of Pp and PA. Now, for any supposed value of Pjp, PA will be 



jvery different place on the earth, and therefore the 

 values ot these integrals will be every where the same. If we 

 denote them by M and N, we have 



r«ATM+A(«-T)N = AM.T+AN(/-T). 

 By the same process we obtain for the vertical force due to the 

 arcpB t/ = 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 t—V will remain very nearly the same, except 

 at considerable depths where the rate of variation of the temper- 

 ature may be different, or the arc AB may be diminished by the 

 absorption of the ethereal waves in their passage to the surface. 

 If we neglect these possible variations of t — t' and add together 



tical force 



actual 



V = C(/-n . . . (2.) 



in which C is the sum of the values of c for the different laminae. 

 If we take account of the variations of t — t', we shall have the 

 actual force equal to the sum of a series of expressions of the 

 form c(t - V) in which both c and t — V will be more or less dif- 

 ferent. It would seem, however, that the changes in the value 



of t 



be 



In fact if the absorption be always a certain fractional amount of 

 the intensity, there will be no change of t — V from this cause. 

 It will only be necessary to regard c as varying. And if the ab- 

 sorption be always the same fractional amount whatever may be 

 the intensity, c and therefore C will have the same value at dif- 

 ferent places. 



The supposition made in the investigation of formula (2), that 

 the variation of temperature is uniform for the extent of the 

 arc 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 

 greater, except in the higher latitudes, than formula (2) would 



Second Series, Vol. IV, No. 10.— July, 1847. 2 



