PROCEEDINGS 



THE ROYAL IRISH ACADEMY. 



1846-7. No. 57. 



November 30th, 1846. (Stated Meeting.) 



REV. HUMPHREY LLOYD, D. D., President, in the 

 Chair. 



Thomas Moore, Esq., having been specially recommended 

 by the Council, was elected an Honorary Member. 



The Rev. William Roberts, F. T. C. D., read a paper on 

 the definite integral 



n" log(l + nsln^) 



It is clear that the only admissible values of n (real) are those 

 of the parameter of an elliptic function of the third kind, to 

 the modulus k, namely, cot-0, — 1 + ^^sin'^^, and — k^sin^O, 

 where k' is the complement of k. The value of the definite 

 integral may, in each of these cases, be expressed by elliptic 

 functions of the first and second kinds, and by the remarkable 



transcendant Y ( \— m — - , ■> • o n 1 by the aid of which func- 

 \J-v/(l — A^sin-^)y ^ 



tions of the third species, with a logarithmic parameter, can be 



VOL. III. 2 c 



