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8 Prof. W. A. Norton on Terrestrial Magnetism* 



Atf(\/l 2 +x 2 )dx. Integrating this between the limits and &, 

 we have, vertical action of GB = At.F(l, k). Whence, vertical 

 action of AB=2A*.F(7 ; k). If k may be taken sensibly the 

 same for different isogeothermal lines, this expression will become 

 2At.¥(l). It is to be supposed, however, that the last particle 

 of GB, which has a sensible action upon the needle at P, is at 

 the same distance from this point whatever may be the distance 

 of AB from it. The value of k will therefore be less, in propor- 

 tion as the distance I is greater. Supposing the most remote par- 

 ticle to be at B, and denoting its distance PB by d, k will be 



equal to \/d 2 — l 2 ^ and the above expression will become 2At. F 



(iyd 2 —l 2 ), or 2At. F'(l). It follows therefore that the entire 

 action of any isogeothermal line AB in the vertical direction upon 

 a needle at P, may be reduced to a single force, proportional to 

 the temperature, and varying from one isogeothermal line to an- 

 other, with the distance PG of this line from the station of the 

 needle. The entire effect of any single lamina of matter will 

 therefore be the same as if the action was confined to the parti- 

 cles lying in the arc GPH ; the effective force of each particle 



being proportional to its temperature, and also a certain function 



of its distance from the needle. 



This being understood, let Fig. 6. 



AB, fig. 6, represent an arc 

 crossing the parallel isogeother- 

 mal lines at right angles, T 

 the mean annual temperature 

 of the earth at p the station of 

 the needle, t and V the mean 

 temperatures at the extreme 

 points A and B which have 

 a sensible action upon the needle, u the difference between the 

 mean temperature at p and at any point w, y the arc pm, a 

 the arc pA, and r the distance Pm. pm or y may be regarded as 

 depending for its value upon Pm, Vp, and Op ; of which Vp and 

 Cp are constant for the same arc. Thus for any one arc, (repre- 

 senting, according to what has been shown, a single lamina,) 



c 



3/=cp(r). If we regard the variation of temperature as uniform 

 for the extent of the arc AB 



V <r(r) 



Thus, temperature at w=T+(*-T) — '• 



Whence, putting v — vertical force due to an element dy at m, 

 and taking the expression for the action of an isogeothermal line, 

 and incorporating the 2 with the constant A, 



