1-24 



Proceedings of the Royal Irish Academy. 



The division into the two classes discussed in § 3 and in § 4 may appear 



artificial ; but, on the whole, I think it the most convenient. It might 



appear simpler and more uatiu'al to separate from the class of equations now 



to be discussed either, (1), those in which the right-hand members are zero 



but A " abnormal," or else, (2), those in which the right-hand members are not 



zero but A is " normal." Of these cases, however, (1) is really very little 



simpler than the more general, and in the case of (2), in attempting to write 



down the solution first, and verify it afterwards, I tried a form which, 



although correct when A is " uonnal," proved to be incorrect in the contrary 



case. 



§ 4"1. Tlie solution ohtained directly from, the eqvMions. 



Proceeding as in § (3"2) we obtain, instead of (42), 

 and, integrating by parts, as with (42), the left-hand member may be written 



0, (46) 



gXi- 



hni^') - tuWi^ + {»io(-D') - <pjky,y 



ly-x 

 iy~x 



p-\f 





f 



Jo 



eHt-nxdf 



eHt-nf{t')dt', ^,,(A) 



(47) 



We will eventually multiply this by dX/A[X), and integrate round the 

 infinite contour, as was done with (43). But, as has been indicated in 

 § (3-1), the argument used of (43) is inapplicable when A is " abnormal." For, 

 if we treat 47j as we did (43;, the coefiticient of dX in that part of the 

 integrand in A which comes from the determinant in the fii'st term will be a 

 fraction whose denominator is A(A), and whose numerator may be of order as 

 high as, or higher than, A (A). I, therefore, proceed to replace this determinant 

 by another, equivalent as regards the solution, which contains no terms of 

 order as high as that of A. 



Eegarding the determinant in question as a fraction whose denominator 

 is -Z7 - A, consider the numerator, viz. : — 



i'Pui^ - ?>uW!« + I*i2(^) - *i2(A)iy, 0i,(A) 



taken at any time t'. 



(48) 



