PEIRCE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNET. 



169 



and at the boundary the surface condition 

 dw s 



w. 



+ /■ 



dt 



+ 



*//(£+ £>---■* (w) 



and which has the given constant value C r on so much of the acy plane 

 as lies within S and the value zero when z is infinite, and if we assign 

 to the function without *S' m where it is not defined, the value zero, then, 

 apart from differences of orientation, all these functions will be alike. 

 If after this we define a function within # by assigning to it within 

 every one of the regions t x , r 2 , t 3 , ■ ■ ■ , the same value as the w func- 

 tion belonging to this region, and give to it in r the common value w s , 

 the function thus determined will be the unique function U described 

 above. 



If after a steady current of intensity ,E/w has been running for some 

 time in the coil of the solenoid under consideration, so that the mag- 

 netic field within the core (which in this case 

 shall be built up, in the manner shown in 

 Figure 59, of filaments of square cross- 

 sections) has everywhere the given constant 

 value //,„ the coil circuit be very suddenly 

 broken, the value of H falls instantly, not 

 only at the outer surface of the prism, but 

 also at the surface of every filament, to zero. 

 Inside every filament 



dH 



dt 



p 

 4~u 



m+W)- ™ 



y 



Figure u'.». 



When t = 0, H = II everywhere within the iron, and when t is in- 

 finite, the field intensity is everywhere zero. According to (I), there- 

 fore, we may consider every filament by itself. 



If we seek a solution of the equation (51) which shall be of the form 

 X YT, where X involves x alone, Y involves y alone, and T is a 

 function of t alone, we shall obtain the expressions 



X = ^l 1 -coscur + ^l 2 -sinour, Y = B x -co i s,fiy+ B*-smfiy, T = e~ AH , 



(52) 

 where 



4 -jj. 



(53) 



