THEORY OF THE DIPPING-NEEDLE. 85 



The simplest case of these equations will be that given 

 by the usual assumption, that the intensity of. magnet- 

 ism is the same after reversion, or that n=l. The 

 four equations then become 



W x (I cot 6 l + t)+2Va cot O l - 2 Ha = ; 



Adding the first pair, and dividing by cot 6 l + cot 2 , 



Wl + ZVa -- ,,=0. 



cot e l + cot 2 



Adding the second pair, and dividing by cot 3 + cot 4 , 



-TFZ + 2Fa 

 Adding these equations, 



cot 0. cot 0. 



Now, considering the vertical force F and the hori- 



zontal force H as resolved parts of the total terrestrial 



7. 

 force acting on the needle, it is seen that -^ is the tri- 



gonometrical tangent of the angle which the direction 

 of the total force makes with the horizon, or is the 

 tan. Dip. Thus we obtain 



