Heterogeneous Elliptic Cylinders at an External Point. 475 



o£ the distance from the centre), and the potentials of hetero- 

 geneous circular cylinders can in the same way he deduced 

 from the other formula in § 4. 



Of course all these results admit of easy verification by 

 direct integration. 



We shall "deduce another simple result from the series for 

 the potential function in § 3. Let us calculate the attraction 

 of the elliptic cylinder at a point on the major axis produced 

 of the section. On the major axis 



and the attraction is 



^ = 



/e=o 



Differentiating the series of § 3 we have 



(l-e°)i fdV\ _1 - (2n + l) 1_ _c^ 



irP \ d r h=o r »=i ' « ! (n f 2$) V 2"' 1 ' »•*•+" 



v<W*=o b' k=i n!<n + 2)I \2/ >'" +1 J 



ab r 1 6 2 1.3c 3 1.3.5 c 4 ] 



-4»-,[fi + i)-(i t ,iy] 



«/> 



— ^ W V2 [( c + r )~ s/{c + r)*— c' 2 ] 



ab 



where f is the distance of the point from the centre of the 

 ellipse. This is the total attraction of an infinite elliptic 

 cylinder at an external point on the major axis of its section, 

 a very well-known result. 



It is also interesting to note that when the cylinder is 

 heterogeneous, the density at any point of the section varying 

 as the inverse focal distance, the attraction at any point on 

 the major axis is similarly expressible in a very simple form. 

 Using the corresponding formula of § 4 we have as before 



