PEIRCE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNET. 165 



(3) IF on Sf^ is a function ( IT^) of z only, such that if n indicates 

 the direction of the external normal to >.% 



»''»+^/C^')*=»' 



(36) 



where k is a given positive constant, and the line integral is to be 

 taken around the perimeter of a right section of Sq made by the plane 

 z — z; and, hence, if 



(4) / / ( -o~ ) '-^"^i taken over so much of the xi/ plane as lies within 



So, is given, then TF is uniquely determined. 



If we assume that two different functions ( W, W) may satisfy all 

 these conditions, and denote their difference by it, 



L (u) = 0, within >So, 



u and dti/dz vanish at all points 

 within Sq, for which z is positively 



infinite, 



u vanishes at all points on the .ri/ 

 plane within aS'„, 



ii on >So satisfies the equation 



Us + ^ 



/G~)--- 



(37) 



Figure 57. 



If we use the space bounded by ^S'^, the x>/ plane, and the plane 

 2; = GO , as a field of volume integration, and denote the whole bound- 

 ary by S; then, since cos (z, a) vanishes on So, and u, cos {x, »), 

 cos (3/, ?i), vanish on the portions of the planes z = 0, z= ^ used as 

 boundaries, (35) yields the equation 



Now ti has the same value at all points on the perimeter (s) of any 

 right section of Sq, so that 



