180 PROCEEDINGS OF THE AMERICAN ACADEMY. 



t 



If the core of the solenoid were made of wire one tenth of a milli- 

 meter in diameter, such as is now in common use in coils intended for 

 loading long telephone circuits, we should have b = 1/200, 7i = 1000, 

 A = 1/lOOUOO, and m would need to satisfy the equation 



J^ (mb) = 100000 mb ■ Jx (mb). (99) 



It is easy to see that the first root of this has a value very nearly 

 equal to 0.0044721, and that the effects of eddy currents would be 

 quite inappreciable. 



Having considered somewhat at length — on the supposition that 

 the induction flux in the air spaces of the core might be neglected — 

 the manner in which a current in the solenoid would decay if the 

 electromotive force were suddenly removed from the circuit without 

 changing the resistance, we may now return to the more general case 

 to which the equations (74) and (76) belong, and remark that in the 

 ideal case where eddy currents are supposed to be absent (68) takes 

 the form 



whence H's = Ho ■ e-^-^^^^'K (101) 



It is clear at the outset that the larger roots, at least, of the two 

 equations (76) and (79) will be very different, since the second mem- 

 ber of (76) soon has a negative coefficient. If then the coefficients of 

 the series (77) could be found, the series (74) and (83) would not re- 

 semble each other in appearance for large values of b and small values 

 of the time. If, however, b is fairly small, as it usually is in practice, 

 we may dismiss all thought of the infinite series, since it is easy to 

 choose the coefficients of two or three terms of the form (73) so that 

 the initial condition shall be satisfied very approximately. In many 

 cases one term suffices. 



Let us consider first the case — already treated in another way — of 

 a square core of 100 square centimeters cross-section, built up of long 

 straight wires 1 millimeter in diameter; so that b = 1/20, n = 100, 

 1/3' = 1.36620 m'b'^, hr = 1000, and the equation for 7)ib has the form 



J. , , 1000.r ^ , , ^ ^ , 



•^^")= l-1.3662U.r -^'^'')- ('«-> 



