1889.] A. Mukhopadhyay — Elliptic Functions and Mean Values. 201 

 Hence the 



sector POQ = ia&sin-l ^ | • 



'Therefore, finally, the whole area common to the two curves is 



For purposes of verification, we may note that, when r = a, this 

 becomes irah, and when r = 6, it reduces to irh^, as are geometrically 

 evident. 



§ 2. Mean value of the Gommon Area. 



Let us now investigate the average area cowimon to an ellipse and 

 a concentric circle of variable radius which always intersects it. 

 Let A be the common area when the radius of the circle is r; then 

 the required mean is given by 



ilr 



fh 



whence 



^ a 



I A dr 



__h 



(a-?OM=J A*- 

 But, from the formula proved in § 1, we have 



Hence, 



where 



X 



A<lr = 2ah P + 2Q, 



b 



