98 Proceedings of the Royal Irish Academy. 



pi C2 



(3) 2a-Z = PJ, - 2/0 - 2 'L= • 



(4) ^a?L = (2P, - Pr) /o + Pi/i - 2 / 



and since Js, that is i,„_i, does not enter into (4), we can eliminate 

 /q and /i from (2), (3), and (4), thus obtaining 



i^^^'^'-f^-'-nV 



I£Jp..^sV-\ 1 



From the two following equations we may determine I3 and I^ in 

 terms of I2 and the Z^. 



(6) sVz = 2/3 + P1/2 + (2P3 - Pf ) A 



+ (3P3-3P,P3 + Pr)/o-2r-^. 



(7) ^a'L = 47; + P,/3 + (2Po - Pi') L + (SP, - 3PiP. + Pi^) 7i 



+ (4P4 - 4P1P3 + 4Pi2p, - 2Po2 - P,') 7,-2 



^5 



y/c^) 



and in general it is not difficult to see that we can express /„, Ii . . . 

 7„,_2, in terms of the Z^; that I^-i cannot be expressed in terms of 

 these integrals owing to the disappearance of the term 2"*"^ m"'~^ in 

 R (z, n), and that we can, in consequence, find a new equation of 

 connexion between the Z^, reducing the number of independent Z, to 

 2m - 2, and that we can express the integrals /,„, /„, + 1 . . . /2m - 3j in 

 terms of the Z^ and the integral I^ _ 1. 



8, "We now turn to the discussion of /,„_i. 



If we write p^ = - where m and v are factor.': of f[z) of the same 



degree so that f'^~) = uv. 



"We find, on differentiating, '2pdp - ■ , when J stands for 



V' 



the Jacobian of ti and v, and is of the 2m- 2 degree in (z). 



/ , s TT ^7 mJdz mJd% 

 (1) Hence 2dp = =^ = 



