Ore — Stability or Instability of Motions of a Perfect Liquid. 67 



the relative velocity in the direction of flow bears to the radial velocity a 

 ratio of order l/.s. 



Thus, as the initial relative velocity in the direction of flow exceeds the 

 relative velocity in a ratio of order cV'V^ the increase in the resultant 

 relative velocity is of order c^7'"V^ 



We may, as with the previous problems, use our results to obtain 

 approximately the kinetic energy of the relative motion. If T be the amount 

 of this energy for unit length of the cylinder, we have 



2r= 



27r Ca 



r (v^ + v"') dr clB 



3 J h 



u -~-rv~ \clrcl\i 

 \h\ dO dr J 



= + 



b {■^v)j, - a (x(,v)a - ^i-d (rv)/dr + dujdQ] drdO. (23) 



The first and second terms vanish since \p is zero along the bounding cylinders. 

 As regards the remaining term, the value of xp is given approximately by (22), 

 and the value of 



du/dd - d{rv)ldr or d^x^jdr- + Ijr . d-^jdr + l/r^ d^x^ldB'' 

 is given accurately, i.e., as far as the first powers of small terms, by an equation 

 of type (8) ; and in this case is, (see (10)), 



{4c'r-* cos c\r-- - b'-) - (4c*r-8 + s^r-^) sin c-{r-^ - b'^) ] sin si Q-{C+ C'r-'')t] . 



(24) 

 Eeplacing the product of two trigonometrical functions by a sum or difference, 



and substituting the critical value of the time, viz. t = c^/{sC'), this may be 



written as the difference of two expressions, thus : 



i {4cV-* sin {sO - c^CjC - c'b--) + (4c*7-« + s-r'-) cos {sB - c'-CIC - c'b-'-) j 



- i (4cV-*sin {2c-r-'+rCIC'-e'b-' - sd) + (4cS'-«+ sV^) cos (2c'r--+ c'C/C- rh"-s6)]. 



(25j 

 Evidently, in multiplying this by (22), in order to find the integral which 

 constitutes the final term of (23), we may neglect the second of these two 

 expressions, owing to its rapid fluctuations with respect to r; and thus, 

 performing the integral with respect to 0, we have the approximate result 



2T ~ TTC^.r- (4c*r-i^ + sh^-") dr = iwe'b-'s-\ (26) 



J 6 



provided c^b~^s''^ is large. 



The initial value of T, obtainable in the same manner or otherwise, is 

 given approximately by 



2 n = TT [ " ! 4c V + s'r-') dr = 4 7rc^/r=. (27) 



Thus the kinetic energy increases from its initial value in a ratio of order 



