Heterogeneous Elliptic Cylinders at an External Point. 4(57 



r being supposed to be greater than the maximum radius 

 vector, i. e. the length of the major axis from S to the 

 remoter vertex of the ellipse. 

 Hence 



2 J ^(l + e cos <j>y & 



_^ Z»+ 2 IT cos n(<l>-0)d<l> 

 2i »(w + 2)' r' l J_ ff (T+^cosTr+" 2, 

 But 



J_ i7 (l + ^cos(W w+2 l j (n + l)J nl'2— 1 (i-^ + | 

 an d f* sin n4>d(j > _ 



J_ w (l + ^COS</)j' lT2 



Hence 



7rP 



logr-S(-l) n . wI („ + 2)1^-1^?; ~iS— 



But „ = ae=CS, where is the centre, so that 



we have finally 



7r/ 2 & »=i H . n ! (?i -f- 2) ! 2" - 1 \r/ 



and 2V is the potential of the elliptic cylinder neglecting an 

 infinite constant. 



4. 



Let us suppose the cylinder to be heterogeneous and any 

 line parallel to the axis to be a line of equal density. Let 

 the density at the point (.?;, y) on the elliptic section be 

 f(x, y) where / is a rational algebraic integral function 

 in x and y. Such a function is also expressible in a series 

 in p and </> of which the typical terms are p^cosqcj) and 

 pP sin </(/>. It will be sufficient for ns to work out the case 

 of these two densities. 



(i.) Suppose the density a = pv cos q$ . 



Then as before 



V = (iff log AP pdpd$ 



the integration is to be carried over the entire area of the 



2 K2 



