60 Proceedings of the Royal Irish Academy. 



and proceed as in Art. 20 to investigate the approximate value of the relative 

 kinetic energy at the critical time. With the notation used therein, 



TIlT 



, I du dw\ , , ,<-,^_. 



where dujdz - div/dr is given by (73), and its approximate value by (74). 

 The approximate value of \p, however, is not now as given by (75), but, as 

 derived from (86), is 



^ = - 2m'{a-r){r-h) 



X {{a + h)r^ + (a + Vfr"- + {a + l){a- + ab + b'^)r + ah(a- + cdi + h'^)} {15k{a + b)]-^ 

 X sin { Icz - mr-b' - m'C'/C] . (88) 



It seems unnecessary to evaluate the approximate expression for T, as it is 

 somewhat complicated ; evidently, however, it is of order 



m^k-- (a - by (a + bf, 

 whereas its initial value is of order 



ni'k-^ (a - b) {a + by. 



The ratio of the increase is thus of the order m* (a^ - b'^y ; and as the ratio 

 of ii} at the critical time to the initial iv"- has been shown to be of the smaller 

 order m^{a + by {a - byk', it is evident that at the critical time vj is order higher 

 than u, exceeding it in a ratio of order [k (a - b)]'^. 



Here, again, by way of verification, we observe that, if we now suppose 

 the ratio b/a to become indefinitely near unity, we revert to the problem of 

 the preceding Chapter for the case in which, in the notation of that Chapter, 

 lb is small ; and it may be verified that under these circumstances the value 

 of w, given by (86) above, agrees with that of v given by (38), Chapter I., 

 and that \p of (88) above agrees with \p of (44), Chapter I. 



Another set of circumstances in which the propagation of the disturbance 

 instanced above might be investigated in some detail is that in which 

 m (a - b) is large, and ka, kh large, but k(a - b) small. But from what 

 precedes it is sufficiently evident that this cannot differ appreciably from 

 the case just referred to of the principal problem of the preceding Chapter. 



There seems no reason to suppose that the possibility of great increase 

 is confined to disturbances of very great or very small wave-length in the 

 direction of flow ; the discussion of this Chapter deals in detail with cases 

 only of one or other of these extreme types, for the reason that for them the 

 formation of numerical estimates is less difficult. 



