122 



Proceedings of the Royal Irish Academy. 



one of them, arising from the second terms of the numerators. The terms of 

 this order in one or both are the same as those in 



A-Vn(XK A-V2i(X>:p, (38) 



Xp being the letter used in (25) to denote the initial value of D^x. Conse- 

 quently, the integrand in the contour integral tends asymptotically to 



X-%, (39) 



and thus the terms depending on x and its derivatives contribute to the 

 integral the value 2iTixp. 



It is important to note that this result would not follow if A were 

 "abnormal"; for then, although there would be no terms in f32), as thus 

 altered, of order m + ?i - 1, the asymptotic value of the integrand is not 

 necessarily (39), and, indeed, is often of higher order. 



Consider, next, the terms in the determinant (in the numerator of the 

 integrand for 2TriD''x) which depend on the initial values of the other 

 unknown, y, and its derivatives, viz.. 



\» 



XP 



>i;(i>) - 012(A) 



B-X 



n-x 



y 



y 



t= 



0n(A) 

 04A) 



(40) 



It should be noted that at least one element in the first column is of 

 order p -v n -\, and in developing this determinant directly this element 

 has to be multiplied by an expression which may be of order n, so that, 

 if n > on, we must adopt some device different from that which was applied 

 to (32). The determinant (40) is, however, equivalent to 



X'',p,,{JD)y, 01, (A) 



A^02,(Z')3/, 022(A) 



(D-A) 



(41) 



t = o 



for this latter is an integral polynomial in I) and A, since it vanishes on 

 replacing I) by A, and the numerators in it and in (40) (which have i? - A as 

 denominator) differ by a determinant which has two columns identical and is 

 therefore zero. 



Consequently, if jj < m, (40) does not affect the initial value D''x; for (41) 

 does not, being of too low an order in A. It has thus been shown that (25) 

 gives the assigned initial values. 



In the very special case in which m is zero, the arbitrarily assigned initial 

 values referred to at the head of this § do not include that of x. 



