104 ON MAGNETISM. 



theory is as follows. In Figure 38, let the magnetic 



Fig. 38. 



pole be defined by prolonging the axis of the magnet 

 till it cuts the earth's surface ; 6 will be the comple- 

 ment of magnetic latitude. The action of the northern 

 or blue pole upon the red end of a needle at P will be 

 represented by 



B (a 2 + Z> 2 - 2ab cos ff)' 1 : 



its resolved part in the horizontal plane at P, towards 

 the pole, will be 



b . sin . (a* + 6 2 - 2ab cos 0)'*; 



the action of the red pole in the same direction will be 



Bb . sin . (a 2 + 6 2 + 2ab cos 0)~* : 



the total horizontal force will be the sum of these two 

 quantities : which, retaining only the first term in the 

 expression when 6 is considered a small quantity, is 



2Bb . sin . a" 2 . 



A similar expression with opposite sign gives the action 

 on the blue pole (the needle being considered to be 



