135, 136] LOGARITHMIC POTENTIAL. 335 



Put 



z = Q tan #, 



dz = 



^c\\ w 



oO) ^= 



2d><? (7 



== -- = > as before. 

 Q Q 



If we had attempted to verify the value 48) of V by direct 

 calculation, we should have found a difficulty in the appearance of 

 a logarithm which would have become infinite when the length of 

 the cylinder became infinite. Nevertheless the attraction is finite, as 

 just shown. It is to be note that all the properties hitherto proved 

 to hold have been for potentials of bodies of finite extent. 



136. Logarithmic Potential. We may state the above result 

 in terms of the following two-dimensional problem. Suppose that 

 on a plane there be distributed a layer of mass in such a way that 

 a point of mass m repels a point of unit mass in the plane with a 



force where r is their distance apart. The potential due to m is 

 V = m log r and it satisfies the differential equation 



ar ar 



+ = 



Similarly, in the case of any mass distributed in the plane, with 

 surface -density /it, an element dm = [jidS produces the potential 

 , and the whole the potential 



51) F = -r f fdwlogr = - I CplogrdS, 



where r is the distance from the repelled point x, y to the repelling 

 dm at a, &, so that 



We may verify by direct differentiation that, at external points, 

 this V satisfies 



dv 



dx 



== ~~ WxJ J 



WEBSTER, Dynamics. 25 



