AND OF A DISK OF SENSIBLE THICKNESS. 675 



If we write 4^' H^P' 



where r, and r 2 are the distances of a given point from the ends of the line, 

 and if the linear density is expressed by 



,, , 



w 



where P< is the zonal harmonic of degree i, then the potential at the given 



point (a, ft) is V = 4<&() p i08), 



where Q t is the zonal harmonic of the second kind, and is of the form 



where -R.-(a) is a rational function of a of (* 1) degrees, and is such that 

 Q { (a.) vanishes when a is infinite, thus : 



5.7 3.5 t , 3\, a+1 5.7 , 5.11 



4 2 J \ \f\rf ._ fi~ _L /I 



-1 4 12 



At a point at a very small distance & from the line, if we write 



_L = log j +log ^ , 



the potential due to the distribution whose linear density is 



is approximately 

 * 



I 



2.3.3 2.3.4.4 



852 



