. [du chv\ , - . ^, 



Orr — Stability or Instability of Motions of a Pe)fect Liquid. 57 



Denoting the relative kinetic energy by T, we have 



TJTT = r [li^ + vr) dr dz, 



d^ d^p\ 



u ^ - vj y- dr dz, 

 dz dr J 



the former integral being taken over the bounding surfaces and A, v denoting 

 the direction cosines of the normal. If the length of pipe included is a 

 multiple of a wave-length, this integral is zero, since over the circular 

 boundaries tp vanishes, and at the two plane ends the values of ^ are 

 identical, and the values of v^i - \w equal, but of opposite signs. Thus, we 

 have only to deal with the second integral, and in it du/dz - dw/dr is seen 

 from (19) to be of the form f {z - Wt, r) ; this, of course, expresses that the 

 vorticity flows with the stream ; by reference to the initial conditions we have 



dujdz - dwjdr = \ ( — -^ — + ^+ — sin m^ (r^ - ¥) — cos m'^{r'^ - ¥) | cos A- (2 - Wt). 



(73) 

 When this is transformed by expressing the product of two trigonometrical 

 functions as a sum or difference, it is readily seen that at the critical time we 

 have, taking into account only the terms which will be most important for 

 integration, 



dii dw . 2mV . onnru /.-T/tv 



^"^'^""^^'''^^^"^ ^^^^^^' ^^ ^ 



The most important term in ^ again is seen from (71) to be 



xP = - 2m'r^l¥ mi[kz - w^l^ - w?C'lC]. (75) 



Thus (72) gives as the average kinetic energy of the relative motion in the 

 disturbance per unit length of pipe 



27rm«^-* r r^ dr or Trm^ (a« - I') {W)-\ (76) 



The corresponding expression initially is approximately 



777?^* (a^ - &*)/4F. (77) 



Thus, the energy in question is increased at the critical time from its initial 

 value in a ratio which is approximately 



4m^ (a* + a?h'' + U)l [ ZW (a^ + 6^)} , (78) 



a ratio which is of the same order of magnitude as that of the value of u' at 



the critical time to the initial value of w^ ; this proves, inter alia, that at the 



critical time the value of %o is of order at any rate not higher than that of u. 



Here again, if we make the further supposition that the ratio of b to a is 



R. I. A. PROO., VOL. XXVII., SECT. A. [8] 



