PEIRCE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNOT. 169 



and at the boundary the surface condition 



Ws + I- 



du 



' s 



dt 



+ 



"='//(^ + |f)"-«. CO) 



and which has the given constant value U^^ on so much of the xy plane 

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

 to the function without 8,^ where it is not defined, the value zero, then, 

 apart ft'om differences of orientation, all these functions will be alike. 

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

 every one of the regions ti, r^, T3, • • • , the same value as the w func- 

 tion belonging to this region, and give to it in Tq the common value w's. 

 the function thus determined will be the unique function U described 

 above. 



If after a steady current of intensity Ejw 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 i/ii, 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 



dt 





y 



(51) 



FlGURK 59. 



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

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

 fore, we may consider ever}^ filament by itself. 



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

 JT- YT, where X involves x alone, F involves y alone, and T' is a 

 function of t alone, we shall obtain the expressions 



X=^i-coscur-l-^2-sina^, V = Bi • cos (3y + B^ ■ sin (3y, T = e~''''^, 



(52) 

 where 



X^ = '-^^p^ ■ (53) 



4 ^(U 



