1889.] A. Mukliopadhyay — Elliptic Functions and Mean Values. 211 



Therefore 



- dr]. 



(l-e^sin^i?)^ 



We have also, when 



r = 



a, 



Therefore, finally, 



i(a-&)r 



--'Ho 



(l-e^EUl^r?) 



and 



r=-L (2F-,r), 



a— 0 



which shows that the average value of the angle between the diameters 

 can be expressed in terms of a complete elliptic integral of the first 

 kind with the eccentricity for modulus. If I be the perimeter of the 

 ellipse, since we have 



we may enunciate the 



THEOREM. The average value of the acute angle of intersec- 

 tion of the diameters of the curvilinear quadrilateral formed by the 

 intersection of an ellipse and a concentric circle of variable radius is 



where a, b are the semi-axes, e the eccentricity, and I the perimeter of 

 the ellipse. 



§ 6. Mean Value of Intercepted Circular Arc. 

 We shall now investigate the average value of the two circular 

 arcs PL and MN intercepted by the ellipse. Since, PR = ri|/, we have 



and 



4 de (a) ~ " 



dv 



