428 CURRENTS AND MAGNETIC SHELLS. 



Thus the potential at a point P of a closed triangular current is 

 proportional, ivithin a constant, to the apparent surface of the triangle 

 which the current encloses, or to the solid angle under which the 

 triangle is seen from the point P. 



If this angle be called w, we have 



ki 

 V = to + const. 



2 , 



The theorem clearly applies to any given quadrilateral; for we 

 may alwayt divide the quadrilateral into two triangles, and suppose 

 that along the diagonal are two equal currents in opposite directions. 

 By this addition nothing is changed in the electrical system, and the 

 given current is transformed into two triangular currents with their 

 positive faces on the same side. The potential is the sum of the 

 two apparent angles of the triangle, or the apparent angle of the 

 quadrilateral. . 



449. POTENTIAL OF ANY CLOSED CIRCUIT. We can draw any 

 surface through the outline of a closed current, and suppose this 

 surface divided by two systems of lines, into any number of quadri- 

 laterals, and of infinitely small triangles with rectilinear sides. If 

 we suppose the contours of each of these elementary figures to be 

 traversed by currents of the same strength, and the same direction 

 as the principal current, we should obtain a system of closed currents 

 which will be equivalent to the given current, since each of the 

 interior lines is traversed by two equal currents in contrary directions, 

 and the only effective portions are those, the whole system of which 

 forms the given current. As all the elementary currents have their 

 faces turned in the same direction, the potential of the system is 

 proportional to the sum of the apparent surfaces of the elementary 

 currents that is to say, to the apparent surface of the proposed 

 current. 



Hence the potential at any point P of any closed current is given, 

 to within a constant, by the solid angle under which the contour of the 

 current is seen from the point P. 



450. EQUIVALENCE OF A CLOSED CURRENT AND OF A MAG- 

 NETIC SHELL. AMPERE'S THEOREM. Let w be the value of the 

 solid angle under which the contour of the current is seen from 

 the point P, then from the preceding theorem 



(3) V = <j) + const. 



