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



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

 which are parallel to the z axis, if F is a function which within Sq 

 satisfies the equation Z ( F) = 0, and if 



(1) Fand dV/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 .ry plane 

 within *Sq, 



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

 the direction of the external normal to *So 



Vs+l-^ + ^-JJ (^+ ^>-%=0, • (44) 



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

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

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

 uniquely determined. 



(IV) Let *So be a closed cylindrical surface which completely surrounds 

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

 (Si, So, Ss, • • • ) the generating lines of which are parallel to those of >So 

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

 plane c = c be denoted by ■% Si, S2, S3, ■ ■ • . Let the portions of the 

 plane z = z within Si, S^, S3, • • • , be denoted hy Ai, A^, A3, ■ ■ • , and 

 the portion within S^ but outside Si, S-., S3, • • • , be denoted by A^. 

 Let Tg, Ti, To, T3, • • • , represent the volumes of the prisms (bounded 

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

 planes c = c are A^, A-^, A^, A3, • • • . 



In the regions T^J, tj, to, rg, • • •, let the scalar function U satisfy 

 the equations 



dU (c'U dH^\ .... 



dU_ (c-U d-U\ 



