1889.] A. MnkhoY>a,dhja.j~ Elliptic Fimdions a7i(l Mean Values. 213 



+ 2a%^ I ^ ^ 



= 7r6 (a-&). 



Therefore, 



o- = 7r6. 



Hence we have the 



THEOREM. If an ellipse is intersected by a concentric circle of 

 variable radius, the average value of the circular arc intercepted is rrb. 

 2Uli July, 1888. 



XI.— Some Applications of Elliptic Functions to Prohlems of Mean Values. 

 (Second Paper). — By AsuTOsn Mokhopadhyay, M. A., F. R A. S., 

 P. R. S. E. 



[Received October 22nd ;— Road Novemlbor 7tli, 1888.] 

 Contents. 



• § 1. Introdtiotioii. 

 §§ 2—5. First case. 



(§ 2). Expression for common volume. 



(§ 3). Expression for the mean value. 



(§ 4). Geometrio interpretation. 



(§ 5). Canonical form for volume. 

 § 6. Second case. 



§ 1. Introdiiction. 

 In my first paper on " Some Applications of Elliptic Functions to 

 Problems of Mean Values," which was read before the Society in 

 August last,* 1 discussed, among other questions, the problem of deter- 

 mining the average area common to an ellipse and a concentric circle 

 of variable radius always intersecting it ; the present paper is devoted 

 to a discussion of the corresponding space-analogue. Given the ellipsoid, 

 2/2 i?2 



a^'^T^ + c^ = ^' (1) 



* See above, pp. 199—213 ; P. A. S. B. (1888), pp. 184-5. For a fall analysis 

 of the present paper, see P. A, S. B, (1888), pp. 207—8. 



