1889.] A. Mukho-padhjuj—ElUptic Functions and Mean Values. 209 



(1 + m ain^ B){\ - &" sin" 0)^ 

 =.P-2R. 



But, from a known formula (*), we have 



R = - (F - P) 



Therefore, 



-iii^=P+-^P-F) = (l+?)p-?F = 2i±iL> 



This shews that the average value of the angle of intersection of the 

 two curves may he expressed in terms of two complete elliptic integrals 

 of the first and third kind. It is interesting to note that the result de- 

 pends simply on the ratio of the axes of the ellipse, viz., if 6 = a\ 

 ■we have 



Hence we may enunciate the 



THEOREM. The average value of the angle of intersection of an 

 ellipse and a concentric circle of variable radius which always intersects 

 it is 



where X is the ratio of the axes ^ = - j, and F, P denote complete ellip- 

 tic integrals of the first and third kind respectively, the modulus being 

 1 - A.S 



(1 — X^) and the parameter ■ 



§ 5. Mean Value of another cLngle. 

 If we join the opposite corners of the curvilinear area formed by 

 the intersection of the circle and the ellipse, the joining lines will evi- 

 dently intersect in the common centre of the two curves ; we shall now 

 proceed to investigate the average value of the acute angle included by 

 these two diameters. 



* See Dr. ScUoemiloh's TUorie des InUgrales et des FoncHons Elliptiques, par 

 Dr. Graiudorge, (1873), p. 63 ; Oayley's EUvtic Functions, p. 15, § 33. 



1+ x° 

 X(l-X) 



