238 THE ELECTRIC TELEGRAPH. 



these forces may be further decomposed into forces acting at 

 right angles to the direction of the needle and parallel to it. 

 Thus, the force a n, the horizontal force of the earth's mag- 

 netism upon the pole, is equivalent to c n at right angles to 

 the needle, combined with b n r parallel to it, while, in the same 

 way, a n is equivalent to the forces e n and d n'. But as the 

 forces b n and e n act parallel to the needle, they can produce 

 no deflection, whilst c n and d n', acting m opposite directions 

 between which the needle is balanced, must be equal to each 

 other. 



The angle a = n n' = a n b = a' n d] 

 by plane trigonometry, 



n d = n' d. cos. a. 

 n' c = n' a, sin. a. 



And since rid = ric, 



n' a' cos. a = n' a, sin. a, 

 whence 



-? = n' a, tan. a. 



cos. a 



Putting, for the sake of clearness, the horizontal component 

 of the magnetic force of the earth, n #=:M, and the strength, 

 or magnetic force of the current, n a =. S, we have 



S = M, tan. a. 



By the same reasoning, when the strength of the current is 

 altered to S', the angle of deflection will be altered to a'. M 

 remains constant, and 



S:S'=M tan. a : M tan. a'; 

 or, 



S : S' tan. a : tan. a'. 



Therefore the magnetic effects or strengths of two currents 

 are proportional to the tangents of the angles through 

 which the magnetic needle is deflected from the magnetic 

 meridian. 



But to compare the values of S and S', measured with dif- 

 ferent rings and needles, we have the proportion 

 S : S' = C tan. a : 0' tan. ', 



