1889.] A. Mukhopadliyay — Elliptic Functions and Mean Values. 207 



tana)= ^5-^ 



Hence, 



a6 tan <o = v^(a^-r«)(r»- 



§ 4. illeaw FaZMe 0/ ^ngrZe 0/ Intersection. 

 Let us now find the mean value of the angle of intersection of an 

 ellipse and a concentric circle of variable radius which always intersects 

 it. Let (0 be the angle of intersection when the radius of the intersect- 

 ing circle is r ; then, if U be the mean value required, we have 



dr 



whence 



(' 



Integrating by parts, we have 



■a-i)n= f . 



ave 



L -'r=b Jr = l 

 = |(a»-r=)(rS-6-) j ' 



» r = a 



rdo>. 

 = 6 



Now, from § 3, we have 



ab tan 



which shews that, when 



r = a, o) = 0, 

 r=&, <o==0, 



go that the integrated part vanishes at both limits. We also easily calcu- 

 late by logarithmic differentiation that 



rfo) a6(a° + -2r') 



~ r(a» + &• - r»)(a» - r=)^ (r' - 6')"^" 



27 



