Orr- — Stability or Instahility of Motions of a Perfect Liquid. SU 



m^ [o? - V-) is large ; and the second term, by replacing the sine by unity, is 

 seen to be of order not higher than the first (and, as a matter of fact, is much 

 smaller). 



And, as with the former set of conditions, the remaining portions of the 

 second term of (65) are small compared with (66), and would be so even if 

 in them the sine or cosine were replaced by unity. 



And, as before, the argument applies when h is zero. 



Thus, again in (65) only the first term need be taken into account; and its 

 approximate value is given by (82). This, then, is to replace the integral in 

 the first term of (29), and, substituting the approximate values of /, K, this 

 term becomes 



- m\a^- r^) (>--&)^(3r^ + 6?'V:)+4r&^+ 2h^) (30«6r)-i. cos [kz - m-lf - m^G'iC] . (85) 



And, in a similar manner, it may be shown that the second term of (29) is 

 replaced by a quantity differing from (85) only in having a, h interchanged in 

 the multiplier of the cosine, and in having a plus sign prefixed. 



Thus, by addition and subsequent division, the equation which replaces 

 (29) leads to the approximate result, 

 It :=- m}{a- r){r - h) 



X { (a + r)(r - &)(3r^ + Qr^-h + 4r6^ + 25^) + (r + h){a - 7')(3r^ + 6r^« + 4ra.^ + 2a^)| 

 X { Ihria" - lf)p cos [kz - ni'lf - m^C'/G] 



■= - 2ni\a-r){r-h) 



X {(a + h)r^ + (a + hyf- + {a + h){a^ + ah + h'')r + ab{a^ + ab-\- If) ] 



X {15r(a + 5)}-icos{A-2-m^&^-m2(7V(7}. (86) 



Here, again, the value of lu cannot be found from this approximate expression 

 for 11. 



When a -T and r -h are not very unequal, this value of it is of order 

 r/i^{a^ -b^Y; and its initial value is of order unity, so that it increases in a 

 ratio of this order. As in the other cases, the initial value of w, however, 

 exceeds the initial value of lo, now in a ratio of order m'^(a + h)/k ; thus, as 

 far as our investigation has gone, we cannot be sure that the disturbance 

 will increase much, unless m}{a~-h"-)~ is large compared with 'ni^(a + h)/k, 

 or m~(a + b)k(a - by is large, k(a - b) itself being known to be small. This 

 condition for a large increase in the disturbance can, of course, be secured ; 

 but, with the relative magnitudes chosen at the beginning of this Art., it 

 appears that at the critical time the value of w is of order greater than u, 

 and that the additional condition just stated for a large increase is unnecessary. 



We may, in fact, suppose that ma is so large, and ka so small, that (86) is 

 valid, except in comparatively small portions of the stream close to the walls. 



