in disks and thin rods 363 



If a is the quantity of fluid in a portion of the plate whose area is unity, 



/+ rt 



= I pdx = Ca 



J -a 



r /- 



\ 2 



in disk. 



The distribution in an infinitely thin disk may be deduced from that in an 

 ellipsoid by making one of the axes infinitely small. It is better however to 

 proceed by the method which we have already employed, only that instead of 

 supposing the line A t PA 2 (Fig. p. 358) to be a double cone, we suppose it to 

 be a double sector cut from the disk. The breadth of this sector is proportional 

 to the distance from P, so that the condition of equilibrium of the repulsions 

 of two corresponding elements whose surface-densities are cr, and a t is 



,r-f)_P 3-n _ - n P 3-n 



whence we find, as before, that if the equation of the edge of the disk is 



then the surface-density at any point is 



a = Cp n -\ 



The quantity of fluid in the disk is found by integrating over the surface 

 of the disk, and is 2 abC 



<? = T=-T- 



Hence if CT O is the mean surface-density, the surface-density at any point 

 is given by the equation 



rt X n 



- 



Thin rod. 



The distribution on an infinitely thin rod is found by considering A^ 

 a rod of uniform section, which leads to the equation 



where \ is the linear density, and if the length of the rod is 2a, and if x is the 

 distance from the middle, and x 2 = a 2 (i p 2 ), the distribution of the linear 

 density is given by A = 2 



The charge of the whole rod is 



Ca Z- = 2, 



