208 CLAUSIUS ON THE WORK PERFORMED 



time, we shall then have 



H = A.W, (5) 



and hence, according to (I) and (la), 



U = A.fY.idw (IT) 



U=A.k,J^Y.^dw. . . . (II«) 



The integrals contained in equations (I), (la), (II), and (II«) 

 are, in most cases which occur in practice, capable of great sim- 

 plification. 



If a part of the surface enclosing the space under considera- 

 tion form at the same time the surface of the conductor, and if, 

 in comparison to the whole quantity of electricity passing 

 through the conductor, we neglecj the small quantity lost in 

 the surrounding air during the current, then in the integration 

 we may entirely neglect this portion of the surface. When, for 

 example, as is usually the case, the conductor is an elongated 

 body traversed longitudinally by the electric current, and when 

 we consider a portion of it situated between two transverse sec- 

 tions, it will only be necessary to effect the integration for these 

 two transverse sections. 



If, further, the form of the conductor at one of the transverse 

 sections be approximately that of a prism or cylinder, so that 

 we may assume that here all particles of electricity move parallel 

 to each other and to the axis, then the force urging them must 

 also have this direction. Let us now place our rectangular sy- 

 stem of coordinate axes, so that the a? axis may be parallel to the 



axis of the conductor, then -— represents the whole urging force, 

 and —^ and ——vanish. From this position it also follows, 



that when the transverse section is taken perpendicular to the 

 axis, V must be constant throughout the same, and we may 

 write, 



/ Y idw = V /idw. 



The integral jidw, — which must be taken negative or positive, 



according as the transverse section is the first or second in refer- 

 ence to the direction of the currcnt,-^represents the whole 



