130 Proceedings of the Royal Irish Academy. 



i 45. Wlien A is "normal", the m + n initio! values may he assigned 



arhif7-aHly. 



In § (3-1) it has been shown that,, when A is '= normal" (25], which is 

 identical with the part of (57) not involTing /^ /,, gives the correct initial 

 arbitrarily assigned values. 



It suffices, then, to prove further, that the part whieli does involve /,, /,, 

 has no effect on these initial values. And this is easily proved ; for, in (61), 

 the part involving the double integrals is initially zero ; and the asymptotic 

 initial value of the integi-and in the final terms is of order not higher than 

 (and in the ease of one, at least, equal to) that of ^''"^'"'^A; consequently, 

 a 2} lies between zero and m - 1, inclusive, the integral is zero. 



§ 46. Failure to determine the number of riidcpcndent constants in the 

 sohiiion ivhcn A is " abnormal.'' 



As already stated, I have been unsuccessful in my attempts to determine 

 the number of independent constants in the complete solution. At first sight 

 of the form given here it might appear easy to do so ; but when one considers 

 how the independent constants do actually occur in the solutions for the 

 several unknowns, as shown by Eouth and by Chrystal, one may be reconciled 

 to the existence of difficulty. The number contained in the value of any one 

 unknown seems almost ob\'ious from the solution given above ; e.g. the 

 number in a; is the order of A diminished by the order of the highest common 

 factor of A and its first minors corresponding to the column associated 

 with X. 



