CASE OF A SINGLE POLE. 



565 



Let us first examine the action which is exerted on a point 

 A (Fig. 123), by a homogeneous magnetic line of density A, situate 

 along the right line XX'. The action of an element MM' or ds, at the 

 distance r from the point A, is equal to . Let us denote by da 

 the angle between the two lines AM and AM', and let fall the two 



x' 



M: M 



Fig. 123. 



perpendiculars MQ upon AM' and AP, or h on XX'. Then from 

 the similar triangles APM and MQM' we have the ratio 



MQ AP 

 MM'~AM' 



or 



rda. h 



from this follows 



Xds_ da_ Ma_ NN' 



7* it" "FT & 



The action of the element ds on the point A is therefore equal 

 to that of the element NN' of the circumference of radius ^, which 

 has the same density A, or to that of the corresponding element 



of the circumference of radius i, where the density is -. If 



the line in question was bounded at the points B and B 15 its 

 action is equal to that of the arc of circle bb v We at once see 

 that this latter action is proportional to the length of the chord, 

 which gives 



A. , . a 2\ . a 

 / = 2ti sm - = sm - . 

 J h 2 2 h 2 



