136 R. S. Brough — Maximum Magnetic Eff*ect of Electromagnets. [June, 



Put «=1 



Then e=(^ + ^\l 



\n n/ E 



and it is required to make 6 a minimum with respect to 8. 



d /9 

 Putting -— = 0, we have 

 d. o 



S3 (8 + 2p) = ^'^'^^g-^^ {^ (A + a) + 4c} 



which equation expresses implicitly the value of S which makes the mag- 

 netic effect a maximum. 



Let us put ^ = /x, then 

 o 



I 



t/: 



A b (A — a) r , , 



^ ^ ^ TT (A + a) + 4c 



TT (1 + 2 /x) S 



This expression for 8 contains /a, itself a function of 8 ; but a very- 

 simple artifice suffices to get over this difficulty. First suppose />t = 0, and 

 solve the equation : the result will be an approximate value of 8, namely, 

 that which it would have, were there no insulating covering to the wire. 



Then employing this approximate value of 8, calculate jw, = -; and re- 

 calculate the value of 8, using this value of /x. 



By repeating this process, which involves very little trouble if logarithms 

 be employed, any desired degree of accuracy may be attained. 



From the above expression for 8 we see that, so long as (x not = 0, the 

 diameter of the wire (without its covering) will always be less than it 

 would be were there no insulating covering. 



The expression for the resistance of the bobbin may be written 

 ^ X b (A — a) r , , , , ") 



and supplying its value for 8*, we find 



1 +2/X 

 from which jf is seen that, so long as /x not = 0, the resistance of the bobbin 



