96 PROPERTIES OF CIRCULAR CURRENTS. 



In the case of x = 0, the component reduces to 



_ 2 jr a" 5 a' 3 "I 



G ~ ~^ + * [(" + ^)f ('2 + j2 )t J ' ' ' ' 



a< 2 representing the radius of the winding of mean action in 

 respect of the centre (729). 



The value of Y x is then zero, and the component Y is of the 

 third order of y. 



740. PARTICULAR CASES. ACTION OF A COIL AT A SMALL 

 DISTANCE. For the applications it is useful to make the calcu- 

 lation in a more complete manner. 



In order to determine the action of a coil on a point near 

 the centre we might develop the expressions (34), but it is more 

 advantageous to treat the problem directly. 



We may in particular consider the component parallel to the axis. 

 The value X x of this component for a circular current of radius r, 

 at a point whose co-ordinates in respect of the centre are x and y 

 is (736) 



u y^d^u y d*u y d*u 1 



l ~ "" L^~^^ + (^^~(2. 4 .6) 2 ^ + 



We have further, with U 2 = 



dx 2 2 r 2 2.4 r 4 2.4.6 r 6 



From this we can deduce the successive differentials of u in 

 respect of x, and it is easily seen that all the series thus obtained 



