390 On the Amount of Magnetism of a Magnetic Circle. 



Let 8 equal the angle the molecule normally forms with the 

 current, and let a equal the angle to which the current de- 

 flects the molecule. Then the power of the current to deflect 

 the molecule will be equal to sin a, and the force tending to 

 draw the molecule back to its normal position will be equal to 



versin a — versin 8= cos S— cos a; 



but since the action of the current on each molecule is equal, 



therefore = will be the same with every molecule, 



sma J 



whatever may be the value of 8. 



Since the action of the current on the iron is proportional 



to the current-strength, 



cos 8 — cos a 



.-. a=a : •> 



sma 



where a is current in amperes and fju a constant. 



The amount of magnetism of the group of molecules is equal 

 to the summation of the different distances through which the 

 molecules are deflected 



C s=0 ° 

 = /jl \ (cos 8 — cos ct)d8. 



«/<S=180° 



The values of cos a corresponding to the different values of 



cos 8 from 0° to 180° in each value of : may be 



sma J 



calculated by the following series : — 



Ci . cos 8— cos a 



bmce a = Lb = — , 



^ sin a 



os a . 

 .'. cos a= cos o sin a, 



.'. cos a = cos 8 v/l— cos 2 a, 



/* 



.\ cosa=cos8— -\/l — 2 < cos 8— -v'l— 2 {cosS... , 

 and the magnetism of a molecule 



= cos 8- cos a = ^ V 1- 2 ^cos 8-^^/fL 2 {cosS.~ ; 



and this series, integrated for all values of 8 between 0° and 

 180°, would give the amount of magnetism (for any current- 

 strength a) of a group of molecules that would in their normal 

 position form a closed magnetic circle. 



If the constant //, is made equal to unity, and a number of 



