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



tion, 5, parallel to the xy plane at any instant, is equal to 1/4 tt times 

 the value at that point, at that instant, of the derivative oi H m & 

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

 tion ahead of ^\ 



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 Hs- If the coil circuit is 

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



Two or three general theorems concerning solutions of differential 

 equations of the form 



'\dx-^ dy-J ~ dz' 



will be helpful to us. 



If V and ir represent any analytic functions of x, y, z, and if L (zr), 

 M{v) represent the adjoint differential expressions 



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

 pressed by the equation, 



fffl' ■ ^ 0-'') - "' ■ ^^^(^')] • dx dy dz = 



9 I 1 [ i' ■ a~ ~ ^'' ■ a~ ) ■ ^0^ (^' n) -dS — I I IV V ■ cos (z, n) • dS ; (33) 

 and it is easy to prove that 

 j j I '■ L(w) dxdydz = g I J r f ^ • cos (x, n) -{- ^- cos (y, n) \dS 



