476 Mr. N. R. Sen on Potentials of Uniform and 



-- =0 on the major axis and 



(l_^)/BV\ 2 « - (2,,)! 1 £_ 



^I~~ W 7»=o _ r-.ii J » : (/« + 1) ! • 2- 1 * r»+ i; 



_ 2 . 2 » - , 1 .:i.r>...(2n-l) /26Y +1 



-r T <•„:!- :' 2.4.6...(2« + 2)W 



Hence the total attraction 

 - 2(** 



= 47T- 

 C 





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

 ellipse. 



9. 



We have so far considered the case of the complete elliptic 

 area. The method of analysis followed here is, however, 

 applicable to the case of an area bounded by two elliptic 

 arcs. As any two arbitrary arcs would make the result 

 cumbrous, we choose here for illustration a very simple case 

 when the result appears in a rather symmetrical form. Let 

 us suppose that the two elliptic arcs have the same focus and 

 their major axes lie along the same line. Let S be the 

 common focus and let the two arcs whose equations are 



/ 



am 



1 + e cus <p 



V 



r i+^cos^' 



intersect at C and D, and let CS make an angle ft with the 

 line from which <fi is measured. P is any point (p, (p) inside 

 the area, and A another point (?% 6) outside at a sufficient 

 distance from the focus. The area is divided into two elliptic 

 sectors by the radii vectors SC and SD, and the potential of 

 the whole area is the sum of the potentials Y x and Y 2 due 

 to the two sectors. We shall suppose the area to be of unit 



