PEIRCE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNET. 167 



(III) If S is a closed cylindrical surface the generating lines of 

 which are parallel to the z axis, if V is a function which within S 

 satisfies the equation L ( V) = 0, and if 



(1) Tand d V/dz vanish at all points within and on >S' for which z 

 is positively infinite, 



(2) V has a given constant value ( V ) at all points on the xy plane 

 within S , 



(3) V on *So is a function ( V s ) of z only, such that, if n indicates 

 the direction of the external normal to <X, 



or 





where / and k are given positive constants, the line integral is to be 

 taken around the perimeter (s) of a right section of H made by the 

 plane z — z, and the double integral Over the section ; then V is 

 uniquely determined. 



(IV) Let *S'o be a closed cylindrical surface which completely surrounds 

 (Figure 58) several other mutually exclusive, closed cylindrical surfaces 

 (*S'i, So, Sz, ■ • • ) the generating lines of which are parallel to those of S a 

 and to the z axis ; and let the intersections of these surfaces with the 

 plane ; = : be denoted by .% s u s 2 , s 8i ■ ■ ■ ■ Let the portions of the 

 plane z = z within S u S 2 , S s , • • • , be denoted by A h A v A 3 , • • • , and 

 the portion within S n but outside S x , S 2 , S 8 , • • ■ , be denoted by A . 

 Let t^ t x , Tg, t s , • ; • • , represent the volumes of the prisms (bounded 

 by the planes z = 0, z = go) of which the cross-sections made by the 

 planes z = z are A , A v A 2 , A 3 , • • • . 



In the regions t , tj, t 2 , t 8 , • • • , let the scalar function U satisfy 

 the equations 



dU _ (&U &TJ\ 



-dz~- g » \ w + df y 



dU_ (c-U c-U\ 

 dz ~ ih \ C.r- + df y 



