Ouii — S lability or Instabilily of Motions of a Perfect Liquid. 53 



order /.•, and subject to the other conditions stated, the vahie of ii will have 

 increased so as to exceed its initial value in a ratio of order rii^jk^. The initial 

 value of V), however, exceeds that of u in a ratio of order m\li, so that the 

 kinetic energy averaged along a definite stream-line can increase in a ratio of 

 order m^//i;^ only. 



In the preceding analysis, no supposition whatever has been made as to 

 the value of & ; and, consequently, by supposing it to diminish indefinitely, the 

 results are applicable to the case of a complete pipe. It is, of course, only for 

 a complete pipe that the steady motion here considered is the same as that 

 which obtains in a viscous liquid. 



We have here, then, an explanation of the observed instability. But the 

 argument for instability, in the case of a disturbance of the type instanced, is 

 weakened by the fact that r^he disturbance does not reach a maximum simul- 

 taneously for different va aes of r ; in fact, the discussion goes to show that, 

 at any particular time, it can be of the order of the maximum possible at the 

 point considered only through a portion of the stream whose thickness is of 

 order hr'lm. 



As affording some check on the accuracy of these results, it may be 

 pointed out that, if we now further suppose that the ratio of 5 to a is made 

 indefinitely near unity, we return to the problem discussed in the preceding 

 chapter for the case in which, in the notation of that chapter, lb is large ; and 

 it is easily verified that, under these suppositions, equations (59), (60) above 

 agree with (38) of the preceding chapter, due allowance being made for the 

 differences of notation. These differences are accounted for to a slight extent 

 by my following Lord Eayleigh ; unfortunately, however, I have introduced 

 another discrepancy by choosing in the initial disturbance in one case the 

 sine-function, and in the other the cosine, of the coordinate measured in the 

 direction of flow. 



Art. 19. The Steady Motion is Stable for Sufficiently Small Initial Distico^bance 



of the Type discussed. 



Moreover, the value of m given by (29) eventually diminishes indefinitely 

 as the time increases. 



Writing the first of the two integrals in (29) in the form 



U sin \kz - U {Cp' + C')\ dp ; 



and integrating by parts, it becomes 

 ?7cos {kz-M(Cp- + C')\ '■ 1 



2ktCp b ~ mc 



cos \kz - kt (Cy + a')\ --{ -) dp. (61) 

 (ipXpJ 



