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



(3) W on S is a function ( W s ) of z only, such that if n indicates 

 the direction of the external normal to 8 Q 



w s+ if( 



en J 



0, 



(36) 



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

 taken around the perimeter of a right section of 8 made by the plane 

 z = z ; and, hence, if 



(4) / / ( -p— ) dS y taken over so much of the xy plane as lies within 



/So, is given, then W is uniquely determined. 



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

 these conditions, and denote their difference by >/, 



L(u)=Q, within >', 



o> 



u and du/dz vanish at all points 

 within S , for which z is positively 

 infinite, 



u vanishes at all points on the xy 

 plane within # , 



u on /So satisfies the equation 



Us + k f(jn 



ds = 0. 



(37) 



Figure 57. 



If we use the space bounded by S , the xy plane, and the plane 

 z = oo , as a field of volume integration, and denote the whole bound- 

 ary by S, then, since cos (z, ?/) vanishes on >%, and u, cos (.r, n), 

 cos(?/, n), vanish on the portions of the planes z = 0, z= x used as 

 boundaries, (35) yields the equation 



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

 right section of S , so that 



I!><^=£"°-' h !t^=-\k^ ^ 



