6 

 H 



Prof. W. A. Norton on Terrestrial Magnetism. 7 



parts which act nearly in the vertical direction upon the needle. 

 If, as we have supposed, the principle of magnetism be analogous 

 in its nature to light and heat, then it must be more or less ab- 

 sorbed in its passage from the lower arcs to the surface ; and 

 there may be a gradual decrease in the extent of the arc which 

 exerts a sensible action upon the needle, as the depth of the arc 

 increases, until at the lower surface of the stratum of sensible ac- 

 tion it becomes reduced to zero. 



Formulas for the horizontal Fig. 4. 



and vertical components of the 

 directive force suited to our pres- 

 ent enquiry, may be easily inves- 

 tigated. Let AB, fig. 4, be an 

 isogeothermal line, and GH an 

 arc of a great circle crossing this line perpendicularly and passing 

 through P the station of the needle. The magnetic intensity o°f 

 the particles of AB is every where the same. Take any particle 

 m and designate the distance Pm, in a right line, by r. Either 

 end of a needle at P will b6 solicited by a force perpendicular to 

 Pm, and in the vertical plane through Pm. This force will be, 

 for different isogeothermal lines, directly proportional to the mag- 

 netic intensity of m, and therefore to its mean annual tempera- 

 ture (/) ; and will, for the same isogeothermal line, vary from one 

 particle to another with the distance r. Its expression will there- 

 fore be of the form A* . <p*{r) ; Ficr 5 

 A being an indeterminate s * 

 constant. Now, let mp, fig, 

 5, represent the great cir- 

 cle immediately below mP 

 in fig. 4, and lying either on 

 the earth's surface or beneath 

 it. We shall have force 

 Pa(due to m) = At . q>(r). The 



: 



component of Pa in the direction of the radius or vertical PC will be 

 equal to Pa . cos aPC =Pa . sin mPR=Pa . q>'(r, h, R) ; R being 

 the radius of the circle, which may be taken equal to the radius 

 of the earth, and h the height Pp of the needle above the circle. 

 We have therefore for the action of m in the direction of the 

 vertical PC, the expression A* . q{r) . <p'(r, h) ; and when the 

 height h is regarded as constant, we have A/. </(r). </'(r) 7 or 

 A/ ./(*•)• To obtain the entire effect in the vertical direction of 

 all the particles in the line GB, fig. 4, let GB be denoted by k } 

 any portion Gm of it by x, and PG by I The action of an ele- 

 mentary portion of GB will have for its expression At .f{r)dx 



* The letters $, /, F, with and without accents, are used in these investigations 



j notions, and are therefore to be read " a function of 



to d '.ate different fu 



