470 



Mr. W. H. L. Russell on 



[Jan. 25, 



January 25, 1883. 



THE PRESIDENT in the Chair. 



The Presents received were laid on the table, and thanks ordered 

 for them. 



The following Paper was read : — 



I . " On certain Definite Integrals/' No. 11. By W. H. L. 

 Russell, F.R.S. Received January 16, 1883. 



The method by which integral (227) was obtained may be thns 

 extended. 



Suppose it was required to obtain 



[a*({ ^±}^±^±^± 



J J \a' + b'x+GX*+e'xZ+ . . ./ 



Put z— a +^+ ca?2 + ea;3 + • : • 



a / + 5'*+cV + eV+ . . .' 



6{x)=a + lx + cx° + ex s + . . ., 



0(aO = a' + &'a5 + c'a; 3 + e'a; 3 + . . ., 



then 0(a?)=z0(a?), and if (a) be any root of the equation 0(») =0, and 

 6', 0", . . . 0', 0", . . ., are the values of 9'x, 6"x when x=a, then we 

 find by differentiating 0(aj)=20(aj) that 



aj =u +u 1 +u 2i ^+u 3r: J-.+ ... 



, TT -r-r rr 200' 0"0 2 



when Uo=«i, Ui==— TJ a = ~ — 



n _30"0 2 _ 90W 600' 2 _fl / "0 3 , 3fl"0 3 



Hence L/(^±^±^^+ : : : ) 



=J^(u i + U 2 , + U 3 . r ^- 2 +...yW . (234). 

 The theorem applies with great facility when/(z)=\/ I of course 



