38 STATIC ELECTRICITY 



The total number of lines of force through a surface 

 is equal to the sum of the numbers proceeding from each 

 element of charge considered separately. For if 7 is one 

 element the number which it sends through an area at distant 



is ? a cos ** where is the angle between the normal to and r. 



r 2 

 The total number sent by all the elements is 



Y^a cos _ v g cos 



*pr- '-jT- 



But 2 g cos = sum of components of intensities along the normal 



T" 



= N, which proves the proposition. 

 We can now state Gauss's theorem thus : The total Jlu.r of force 





through a closed surface is equal to 4-rr X charge within* or i\ ct/na/ 

 to the number of lines of force sent out from that r //*/;;_ 



Fluid displacement tubes used to prove the properties 

 of tubes of force. The properties of tubes of f'onv may be 

 deduced from the properties of "tubes of displacement" in an 

 incompressible fluid, and as these tubes of displacement form a 

 valuable symbolic representation of the tubes of force in magnetic 

 and to some extent in electric systems, we shall here add this 

 mode of proof. 



Imagine space to be filled with an incompressible fluid with 

 "sources" at various points at which fresh fluid can be introduced. 

 and "sinks'" at other points at which fluid can be withdrawn. 

 If a small volume v of fluid be introduced at any point O, then to 

 make room for it an equal volume r must be pushed out through 

 every surface surrounding O. Draw a sphere round O, Fig. '352, 

 with radius OA = r, and let the fluid originally lying in its surface 

 be pushed out to the concentric spherical surface with radius 

 OB = r + d. The volume which has passed through the inner 

 surface is equal to that contained between the two surface*. Since 



