162 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



section (2aX2a) built up uniformly (Figures 59 and 60) of a large 

 number of varnished filaments of square cross-section (c X c), or else 

 consisting of a bundle of infinitely long straight wires. The axis of 

 the prism shall be the z axis, and the .r and ^ axes shall be parallel 

 to faces of the prism. The electric resistance of the solenoid per centi- 

 meter of its length shall be w, the constant applied electromotive force 

 per centimeter of the length of the prism shall be £J, and the intensity 

 of the current in the coil shall be C Within the core, the magnetic 

 field (ff) will have the direction of the ;:; axis, and if q is the current 



flux at any place 



4:Trq = Cm\H, (27) 



or 4 -n-q^ = -g— , 4 TT^y = — — , 4 Trq^ = 0. 



Within any filament of iron in the core, H satisfies the equation 



dH p fc-H c 



dt 4:TTjJi 



m-w)' 



where p is the specific resistance of the iron and fi is its permeability, 

 which for the present purpose shall be regarded as having a fixed 

 value. 



When there are no Foucault currents in the core, the intensity (H) of 

 the magnetic field within has at every point the boundary value Hs 

 or 4 TT JVC, but if positively directed eddy currents exist, H may be 

 greater at inside points than at the surface. We need not distinguish 

 between the flux p through the turns of the coil per centimeter of its 



length, and N times the induction flux Mil Hdxdy through the 

 core, so that we may write 



or by virtue of (28), 



E = 



4:7rN 



-mm-W)'-^- <-) 



where the integration extends over a cross-section of the core. 



The vector H is always perpendicular to its curl, and the intensity 

 of the component of the current at any point in the iron, in any direc- 



