TERRESTRIAL MAGNETIC INDUCTION. 293 



Hcosa situate in the plane which it describes; the value of the 

 resultant of these two forces is 



and the needle, in its position of equilibrium, makes with the hori- 

 zontal an angle /, defined by the equation 



Hcosa 

 cot i = - = cotlcosa. 



The angle * is the apparent inclination in the vertical plane which 

 is at an angle a with the magnetic meridian. If we have a = - , it 



7T 2 



follows that /=- and the needle is vertical. If the angle a changes 

 by - , we have cos ( a- J = +sina, and the value of the new in- 

 clination is 



cot** = + cot I sin a. 



From these two equations we have, 



COt 2 / + COt 2 /' = COt 2 !, 



a formula frequently used in determining the inclination. 



If the needle is loaded with an accessory weight, or, what is the 

 same thing, if the axis of rotation does not pass through the centre 

 of gravity, the direction of the equilibrium is modified. 



Suppose, as a particular case, that a weight /, at a distance d 

 from the axis of rotation, keeps the needle horizontal in a certain 

 plane ; the moment of the magnetic couple is reduced then to the 

 moment of the vertical component, and the condition of equilibrium 

 is 



This condition is independent of the azimuth of the vertical plane in 

 which the needle moves ; the counterpoise which makes the needle 

 horizontal in one plane, would make it horizontal in all planes. 

 Hence, placing a needle on a vertical pivot, we may counterpoise it 

 so that it is always horizontal ; but the weight of the counterpoise 

 depends on the vertical component, and this ought to be modified 

 if we wish to use the needle in other latitudes. 



