1877.] E. S. Brough — 2£aximum Magnetic Effect of Electromagnets. 135 



circular bobbins by simply putting the length of the sides equal to nothing 



in the various expressions. 



Let Y = the magnetic effect of the bobbin 



E. = „ resistance „ „ 



S = „ external resistance. 



E = e. m. f . of the battery. 



and n = the number of convolutions. 



Then ( Jacobi and Dub) 



n E 

 Y = >' 



R + S 



and the problem is to make Y a maximum,* treating the diameter of the 

 wire with which the bobbin is wound as the independent variable, of which 

 n and R are known functions. 



Let A = the outer diameter of the circular ends 

 a = „ inner „ 

 b = „ length of the bobbin 



c = „ „ „ straight sides between the circular ends 

 8 = „ diameter of the wire 



p = ,, radial thickness of the insulating covering 

 and L = the length of the wire on the bobbin 

 b(A — a) 



for each wire being allowed a square, the length of whose sides is equal to 

 the diameter of the covered wire : and 



^ (A — a) r ^(A + a) ") 



2(8 + 2p)4 2 "^^'J 



But 



where X is the specific resistance of the conducting material of the ware, i. e. 

 the resistance between oj)posite faces of the unit cube of the conducting 

 material. 



Therefore 



L = 



R = 



X b (A — a) 



TT 8^ (8 + 2p) 

 We had 



nE 



-^-^4^(A + a)+4e} 



Y = 



R + S 



* The force exerted by a coil on a steel magnet is proportional to Y, whereas the 

 force exerted on a soft iron armature is proportional to Y^, but whatever value of the 

 variable makes Y a maximum, will also make Y^ a maximum, so the one solution meets 

 both cases. 



