S6 Proceedings of the Royal Irish Academy. 



denominator of (28) exceeds Ic everywhere along the line of integration, and, 



having regard to the actual form of 0, that the modulus of d (fj) is less than 



Cfj,'', where C, h are constants. From this we can readily deduce that (28) is 



less than 



a constant x a power of t x g-(6-<'+y)'^/-i<^='. (29) 



But this diminiahes indefinitely compared with t ; and similarly for the integral 

 along the other line : hence, initially, all the time-derivates of (26) are zero. 



A few words of caution seem desirable here. If U denotes the expression 

 (26), the argument that dU/dt, for example, is initially zero, is not that it 

 diminishes indefinitely with t, but that (Ut - U^jt does so ; and so on. 



Discontinuity in i//' causes a similar difficulty initially, and it may be 

 surmounted in a similar fashion. 



Art. 10. Reference to another Class of Physical Problems to which Similar 



Expansions apply. 



I have considered also to some extent the appKcation of the expansions 

 to problems of the type which originally suggested them (see Art. 1, Part I. of 

 the former paper), but for wliieh the metliod is not well suited. In such cases, 

 in the expansion which now corresponds to (5), we must continue the series of 

 terms relating to a as far as those containing ;//"'"''(«), x"'"''(a), and change 

 " of index (p - 1) and (^J - 2)" into " of index <}: 0" ; similarly, mutatis mutandis, 

 for h : usually, q=p. 



Art. 11. Beference to the corresponding Modification of the Trigonometrical 



Integral Expansion. 



The manner in which the integral theorems of Part I., Art. 1, of the 

 former paper should be modified like the sum theorem so as to obtain an 

 expansion satisfying equations analogous to (1), (2) wiU, I think, be obvious. 

 The exponential notation is more convenient than the trigometrical in which 

 the integral theorems were first stated. And it seems on the whole preferable 

 to add the contour integral (3)* to the contour integral in (11)* and take half 

 the sum. When the notation is then changed to the exponential, the integral 

 theorem is exhibited as a limiting form of (5) of the present paper obtained 

 by making h infinite and a zero. 



It should also be obvious that terms analogous to (6), but whose form it 

 seems unnecessary to state expheitly, may be added. 



The application to physical problems analogous to those just discussed 

 presents no further difficulty. 



* These numbers refer lo ihu iormer paiier. 



