118 Dielectrics and Inductive Capacity [CH. v 



any direction is equal to P multiplied by the cosine of the angle between 

 this direction and that of the lines of force, it is clear that the polarisation 

 may be regarded as a vector, of which the direction is that of the lines of 

 force, and of which the magnitude is P. 



The polarisation having been seen to be a vector, we may speak of its 

 components /, g, h. Clearly / is the number of tubes per unit area which 

 cross a plane perpendicular to the axis of x t and so on. 



The result just obtained may be expressed analytically by the equations 



129. The polarisation P being measured by the aggregate strength of 

 tubes per unit area of cross-section, it follows that if &> is the cross-section 

 at any point of a tube of strength 6, we have e = o>P. Now we have defined 

 the strength of a tube of force as being equal to the charge at its positive 

 end, so that by definition the strength e of a tube does not vary from point 

 to point of the tube. Thus the product &>P is constant along a tube, or 

 wKR is constant along a tube, replacing the result that wR is constant 

 in air ( 56). 



The value of the product wP at any point of a tube, being equal to 

 - , depends only on the physical conditions prevailing at the point 0. 



It is, however, known to be equal to the charge at the positive end of the 

 tube. Hence it must also, from symmetry, be equal to minus the charge at 

 the negative end of the tube. Thus the charges at the two ends of a tube, 

 whether in the same or in different dielectrics, will be equal and opposite, 

 and the numerical value of either is the strength of the tube. 



GAUSS' THEOREM. 



130. Let S be any closed surface, and let e be the angle between the 

 direction of the outward normal to any element of surface dS and the direction 

 of the lines of force at the element. The aggregate strength of the tubes of 

 force which cross the element of area dS is P cos e dS, and the integral 



PcosedS, 



which may be called the surface integral of normal polarisation, will measure 

 the aggregate strength of all the tubes which cross the surface S, the strength 

 of a tube being estimated as positive when it crosses the surface from inside 

 to outside, and as negative when it crosses in the reverse direction. 



A tube which enters the surface from outside, and which, after crossing 



