COIL WITH A SOFT IRON CORE. 535 



If the radius of the core is the only variable, and we wish to 

 arrange it so that the coefficient of mutual induction is a maximum, 

 the partial differential of M in respect of x should be null ; it 

 follows that 



I27T/& 



or sensibly 3^=27, since the coefficient of magnetisation k is 

 very great. 



If the external radius z is given, as well as the way in which 

 the wires are coiled, and we wish to arrange x and y so that 



DM DM 



M is a maximum, making the partial differentials - and -^~ 



<te Dj/ 



equal to zero, we shall find in like manner 



34 



Lastly, the coefficient of self-induction of a coil which has the 

 external radius z, and the internal radius y, and which includes 

 a soft iron core of radius x, will be expressed by 



(55) L = ** n \l(z -y? z* + zy +/ + 1 2**** 



The problem we have here discussed is that which occurs in 

 the construction of induction coils. 



555. ANNULAR COILS. In a coil formed by regularly winding 

 a wire on a circular ring, the coefficient g (496) becomes 



Suppose that a second wire is coiled n' times round the first in 

 any way whatever; the coefficient of mutual induction will be 



(56) 



