8o PROPERTIES OF CIRCULAR CURRENTS. 



The quantity a 2 represents the radius of the winding of mean action 

 in respect of the centre. ^ 



Neglecting fourth powers of the ratios - and - , we have 

 , a a 



simply 



730. THE MOST SUITABLE DIMENSIONS FOR A RECTANGULAR 

 CHANNEL.* The length of the wire being given, as well as the radius 

 a' of the core, we may investigate the dimensions of the rectangular 

 section which gives the maximum action at the centre of the coil for 

 a uniform coiling ; the problem then is to choose a" and b so that G ? 

 is a maximum, these two quantities satisfying equation (2). 



Instead of solving this problem by the ordinary method of 

 maxima, it may be arrived at more simply by the following con- 

 siderations. 



It is clear that if the maximum is obtained, the mean specific 

 action of the outer layer is the same as the mean specific action of 

 the lateral layer, and there is no advantage in transferring the turns 

 from one region to the other. 



But from equation (n) the value of the total action of the layer 

 of radius a" at the centre is 



the length of the wire which forms it being 2-n-a" x 2n l b = $Tra"n l b, the 

 specific action is equal to 



i 



A cylinder of the lateral layer comprised within the radii r^ 

 and r + dr, contains a number n^dr of windings; the action of this 

 layer at the centre is accordingly 



* W. WEBER. Galvanomelrie. Abh. der K. Gesellsch der Wiss. zu Gottingen. 

 Vol. x. 1861-62. 



