Orr — Slahilifif or Instability of Motions of a Feifect Liquid. 49 

 and neglecting the value at the lower limit, we obtain 



keos \77ir-mh+kz-Jd(Cr''+C")} +(m-2Ctkr) &m\mr-mh+kz-M{CY+C')\ 

 "" k' + (m-2Ctkry 



(38) 



I think it desirable to examine more carefully the validity of the 

 approximations by which this value for (31) has been obtained, and to 

 specify more fully conditions under which it may be used. It seems more 

 convenient to consider the steps in the reverse order. 



In the first place, when we come to take account of the second integral 

 in (29), it will be found that the term in (38) which involves the sine is 

 cancelled ; and it is therefore a condition that 



cos {mr - onb + kz - kt {Gr^ + C) \ 

 is not too small a fraction. 



The substitution of (38) for (37) is evidently justifiable if e^ (»■-*) is large. 



As regards the replacing of (36) by (37), if we denote by TJ the amount 

 by which the integral in (36), considered as an indefinite integral, falls short 

 of the approximate value substituted for it in (37), we evidently have 



_f^ dU - 2kCt sin {mp -mh + kz-h Wf) 

 ^ 'l^^ WVim^^Wtkpf 



+ ^kCt {m-2Ctkf,) \k cob (mp-mh+kz-k Wt) + {m-Wtkp) sin (mp-mh+kz-k Wt)] 



{k^ 4- {m-2CtkpfY~ 



(39) 



The first term on the right is numerically less than 2Ctlk, and the 



3 



second than 4:kCt(m-'2Ctkp)/{k^ i-('ni-2Ctkpy)-, and this again than 

 4:kCt/{k^ + (m-2Ctkpy\, and d fortiori less than 4:Ctlk. Thus by integra- 

 tion the value of U is certainly less than 6Cte'"'/k\ By using (38) when 

 m - 2Ctkr is small compared with m, and of order not greater than k, 

 TJ has been neglected in comparison with e^'^jk, which is certainly legitimate 

 if Gtlk is small, i.e., if mjU'-r is small. 



Again, at such a time, the ratio of the difference between (35) and (36) 

 to the final result, (38), is of the order of the ratio of 



r 



{r\ - p§) ek? cos {mp ~mb-\-kz-k Wt) dp to rh^^'/k. ( 40) 



b 



This integral is less than 



^\ri-p^)e^^dp; (41) 



b 



or, integrating by parts, than 



A;- j_(,J_^,J),^6+rieV^,7^j; (42) 



R.I. A. PEOC, VOL. XXVII., SECT. A. [7] 



