Heterogeneous Elliptic Cylinders at an External Point. 479 

 Now 



+ cos ( ?? — iw <j) — ?i6) | <i(/> 



and 



j: 



^cos {n + mcj) — nd)d(j) 



, lVl+mo a- cos n0 — (?i + m) sin /*# . , 

 /r + {ji + m )- 

 and 



DS 



# 2 + (n — m) 



, lV ,-, iio « cos t?# — (71 — m) sin nd . . 

 = { — L) I. -— — - — — (- — — sinh/c7r. 



Hence 



V &, _ 2*A 2 



/ 2 





-2 $(-l) n+m 



7i — l ?;j=0 



Tccos 7i# — (n + m ) sin ??( 



« 2 + (?i + m) 2 

 /e cos ?i# — (n — ni) sin n^ -1 



/c 2 + (?i — m-)" 



J n(w + 2) W * 



If we put a: = 0, the cylinder becomes homogeneous and the 

 present series in this limiting case degenerates into the series 

 o£ § 3. Moreover, the case of the density p*e K< P can be easily 

 worked out in a similar manner, and putting Ik (7= */ — 1) 

 for k we can deduce the formulae of § 4. Thus this form 

 appears to embody in itself all the preceding different cases. 



My best thanks are due to Dr. Ganesh Prasad for his kind 

 help and encouragement, and to my friend Mr. S. N. Bose 

 for his encouragement and useful criticism. 



Universitv College of Science, 

 Calcutta. 



