PEIRCE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNET. 163 



tion, s, parallel to the xy plane at any instant, is equal to 1/4 -n- times 

 the value at that point, at that instant, of the derivative of H in a 

 direction parallel to the ,vy plane, and 90° in counter clockwise rota- 

 tion ahead of s. 



Along any curve in the iron parallel to the xy plane, H must be 

 constant if there is no flow of electricity across the curve. At every 

 instant, therefore, the value of H at the boundary common to any two 

 filaments must be everywhere equal to H s . If the coil circuit is 

 broken, // must be constantly zero at the surface of every filament. 



Two or three general theorems concerning solutions of differential 

 equations of the form 



/aV d*w\ dw 



g \dx 2 + df) ~Tz' 



will be helpful to us. 



If v and w represent any analytic functions of .r, y, z, and if L (w), 

 M(v) represent the adjoint differential expressions 



d 2 w d 2 w dw . , 



9'W + g 'W~te> (31) 



d 2 o d 2 v do ,„ . 



Vw + 0'df + Fz> (32) 



the corresponding form of the generalized Green's Theorem may be ex- 

 pressed by the equation, 



fffb ■ L (ir) - w ■ M(v)] ■ dx dy dz = 



g ff\ v ^- w Tx)' G0S ^ n) ' dS+ 



g I J f v • > w ■ — J • cos (y, n) ■ dS —J I w v ■ cos (z, n) • dS; (33) 



and it is easy to prove that 

 III' £(«0 dxdydz — g ' [ J o I =— • cos (x, n) + ^— ■ cos (y, ri) \dS 



