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



we have 



Similarly 



-fi = V (,r-/J, O log r- 2 S ( - lj» +1 ^Trfff (~) cos n6. 



The potential of the complete area V = V 1 + V 2 . 



It should be observed that the quantities /, I', /3, e, e 

 are not all independent ; in fact ft is determined by the 

 equation 



I _ V__ 



l + ecos/3 " 1— £'cos/3* 



When the point A lies within the limiting circle an analysis 

 on the same lines is possible if we divide the elliptic area into 

 two parts (by a circle with S as centre and SA as radius) in 

 which two separate logarithmic expansions would apply. In 

 the most general case of two arbitrary elliptic arcs the area 

 may be looked upon as the sum of two elliptic segments each 

 of which is the difference of an elliptic sector and a triangle. 

 In the preceding analysis we have virtually given the potential 

 of an elliptic sector and the potential of a triangle is known. 

 But as the results in all these cases are not simple or sym- 

 metrical, it is unnecessary to deal with them here. 



10. 



In this connexion we may also study the potential of the 

 complete cylinder when the density is an exponential function 

 of the vectorial angle cf>. As will be seen below, this may be 

 considered as a generalization of the preceding eases. The 

 method of analysis followed would be exactly similar. 



Suppose o- = e K ®. 



Then 



y = zsf**** [ ,o « r -ii(rr wBn »-^>* rf * 



2 J_ ;r (lH-ecos0) 2 ° gr % n(n + 2)r n J_ jr (1 + e cos <£)"+* 



