376 Permanent Magnetism [CH. xi 



The theorem that (JNdS = 0, 



where the integration extends over a closed surface, may now be stated in 

 the form that the number of tubes which enter any closed surface is equal to 

 the number which leave it. This is true no matter where the surface is 

 situated, so that we see that tubes of induction can have no beginning 

 or ending. 



437. Let us take any closed circuit s in space, and let n be the number 

 of tubes of induction which pass through this circuit in a specified direction. 



Then n will also be the number of tubes which cut any area whatever 

 which is bounded by the circuit s. If $ is any such area, this number is 



known to be llNclS, where the integration is taken over the area S, so that 



-SI 



The number n, however, depends only on the position of the curve s by 

 which the area S is bounded, so that it must be possible to express n in a 

 form which depends only on the position of the curve s, and not on the area S. 



In other words, it must be possible to replace I iNdS by an expression which 



depends only on the boundary of the area s. This we are enabled to do by a 

 theorem due to Stokes. 



STOKES' THEOREM. 



438. THEOREM. If X, Y, Z are continuous functions of position in space, 

 then 



[( v dx j v dy ( gdz 

 ds 



O <7 >-\ ~\7\ ,'-\ TT ^\ ^. ,3 ~\T 3 V\ \ 



oA \\ iv*"- \\ ( iL/7^ /'^fi^^ 



where the line integral is taken round any closed curve in space, and the sur- 

 face integral is taken over any area (or shell) bounded by the contour. 



Here I, m, n are the direction-cosines of the normal to the surface. A 

 rule is needed to fix the direction in which the normal is to be drawn. The 

 following is perhaps the simplest. Imagine the shell turned about in space 

 so that the tangent plane at any point P is parallel to the plane of ocy, and 

 so that the direction in which the line integral is taken round the contour is 

 the same as that of turning from the axis of x to the axis of y. Then 

 the normal at P must be supposed drawn in the direction of the positive 

 axis of z. 



