PEIRCE. — CHANGES IN INDUCTANCES OF ELECTRIC CIRCUITS. 577 



within the comparatively short range from to l\ unity may be repre- 

 sented with sufficient accuracy by a few terms (sometimes two) of the 

 form 



Zi • Jo(nhi-) + L2 • Jo{nhr) + Lz ■ Joim^r) + . . . = ^A • M'^h^), (49) 



so that // =. 74 + (//„ - /I J 2A • e-^^' • A (^>cr) (50) 



gives the value of the magnetic field at the time t at any desired point 

 in the wire in question, and, therefore, at any desired point in any 

 other wire of the core. 



w = '^^'Lf"^ ' 2^' • '"'^"' • -^ (»"■). (51) 



and if this be integrated over the cross-section of a wire and divided 



by Trb^, the result, 



will represent the average value in the whole core, at the time t, of the 

 time rate of change of the magnetic field. An example will best show 

 the meaning of these rather intractable expressions. 



Suppose the core of a long solenoid of square cross-section, ten cen- 

 timeters on a side, to be built up of straight, round iron rods one mil- 

 limeter in diameter placed close together; then h = 0.2146, b = 0.05, 

 w = 100. If the resistance of the solenoid coil per centimeter of its 

 length is ^g of an ohm, the specific resistance of the iron 9950 abs- 

 ohms, the number of turns of wire per centimeter of the solenoid 10, 

 and the value of the permeability of the iron 100, then mb = x satisfies 

 the equation 



/o (^') • (1 - 1.3666 o'^) = 1000 X ■ Ji {x), (53) 



and the first root x = 0.04465 will suffice, for m = 0.8930 ; and 

 Jo(0.8930^-) differs from unity by less than one tenth of one per cent 

 over the whole range from r = to r = 6, and from (50) 



^~^°° - ^-6-315^ . J,(mr), (54) 



^0 — ^x 



very approximately. In the case of a core of the same cross-sectional 

 area (0.7854 J.), and the same permeability, but wholly without eddy 

 currents, it is easy to show that 



ff=B^ + (ff,-HJ-e-^, (55) 



VOL. XLVI. — 37 



