166 A. Mukhopadhyay — On some Definite Integrals. [Mat, 



wliere a, b are the semi-axes and 6 the angle between the major axes of 

 the ellipses. 



3. Noting that when z = I 



log z log r-— ^ = 0, 

 I -r z 



it is shewn that 



I ' tan-i } r^^ -T^^ d9 



-1 ^ 2^ 1 > 



*^^ Jn^si^^s 



1 - z C z^ z^ > 



= log.log^ + 2 ^^+35+p+ ]■ 



4. The average value of the curvilinear area common to the two 



curves is 



4<ab 

 . u 



IT 



where 



, h, a-h ^\ 1 /&\2'^-l 



^i = log - log + 'i } 77; 7—?, I - I 



* a ^ a-^ b / {2n - 1)~ \aj 



5, Remembering that 



■1 a sin 6 



tan"^ -cl6 



i + a COS o 



f 



it is shewn from (3) that if 



P = a2 (1 - a2) COS 29 + 2a (1 + a2) cos Q + a^ (a^ + 3) 

 Q = a2 (1 - a2) COS 29 + 2a (1 + a2) COS 9 4(1 + 3a2) 



we have 



»/o 



tan-1 $ ^ cot 9 J cZ9 = 2 log a log ^ ?. 



( C^ > i + a 



Dve integrals vi 



c/o 



o 

 The connection of the above integrals with Schaelfer's integral 



\ {x) = — \ - log (1 - a) da. 



is pointed out. (See Grelle, t. XXX, 277—295). 



Pandit Haraprasad Sbastri read a short account of an old gun 

 recently dug up at False Point. 



On the completion of the light house at False Point, about 1838, 

 three guns seem to have been removed fx'ora the Old Fort at Cuttack 

 and buried in the ground there for the purpose of setting up the rigging 



