206 A. Mukliopadhyay — Elliptio Functions and Mean Values. [No. 2, 



where 



(l-e^sinH)'' di 



so that Ej is the complete elliptic integral of the second kind with the 

 eccentricity for modulus. Therefore, 



If Z be the perimeter of the ellipse, we have 



Z = 4aJ^2 (l-e2sin2^)^ d^=4a^i. 



Hence, finally, we have the 



THEOREM. The average area common to an ellipse and a con- 

 centric circle of variable radius which always intersects it is 



where a, b are the semiaxes, and I the perimeter of the ellipse. 



§ 3. Expression for Angle of Intersection. 

 If (0 be the angle of intersection of the ellipse and the circle, its 

 value may be easily calculated as follows. The coordinates of the point 

 of intersection P being a, jS, we have 



The tangent to the circle at P is 



ax (3y — r^, 

 and the tangent to the ellipse is 



ax Sy_ 



whence we have 



