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



give it, (that is, supposing C to be determined 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 P<2, 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"(/). Hence, that of all the isogeother- 

 mal lines, or of the whole acting surface., will be reduced to that 

 of the single arc 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 

 by an equal diminution to- 

 wards the north. Hence, de- 

 signating the arc pm, fig. 7, 

 by y, and the distance Vm by 

 r, the expression for the hori- 

 zontal force due to this arc is 



fdh^fA'T . F"(r)tfy = A'T/F"(r>fa(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 force 



H=C'T . . . (3.) 



This is the expression for the entire effect of a single lamina. 

 For different laminae 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) answering 

 to the different laminae, 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 re- 

 garded as truly constant for all places. 



