Orr — Stability or Instability of Motions of a Perfect Liquid. 31 



% = d\\jldy, v~- dxp/dx, where evidently \p is given by the equation 

 2l\p sinh Ih 

 {P + mO B 



sinh/5 Q,o^{lx +{m-l^t)y] - sinh l(b-y) cos Ix - sinh ly cos [Ix + (m-l^t)!)} 

 -"* P + (m - i(5ty 



sinh lb cos { Ix- (m + l(5t) y } - sinh I (b-y) cos Ix - sinh ^2/ cos { fo - (m + ^jS^ & } 



(44) 

 If T be the average energy of the relative motion per unit length of pipe 



4r,J = 



277// 



rb 



the former integral being taken over the bounding surfaces. This integral 

 is zero since xp vanishes at the fixed planes y = 0, y = b, and since at the 

 planes x = 0, x = 2-)t/1, the values of \p are identical, and the values of 

 d\p/dn numerically equal, but of opposite signs. Thus we have only to deal 

 with the final integral above wherein 



V^ = Jj - — - — - sin l[x- [5ty) sm my. 



If we retain only the first of the two fractions in the value of \p, as given 

 by (44), we have, on integration with respect to x, 



SP sinh lb T 

 (P + m^Y B'^ 



^ r*sinhZ& sin^ my -Binhl {b-y) sinlfity sinmy- sinh ly sml(5t(b-y)&in7n{b-y) 



lo^ P+{m-l[ity ^• 



(46) 



At the critical time at which m - l[it is zero, we obtain on integration, 



_ (P + 7n^y \ 4:W? tanh ^Ib 



^ ~1.W~ I ~ FT4^ ilb 



while originally To = £^b {P + 7rP)/8P. Thus, m/l and mb being each large, 

 the ratio of increase is great whatever be the value of lb; its approximate 

 values in the two extreme cases of lb great and lb small are respectively 

 my2P and m^by24. 



These results are not substantially affected by conditions which may be 

 prescribed at the ends of the stream, if the distance between them is large 

 compared with tt/I. For example, if the end-conditions be those of Art. 6, the 

 additional terms Ui, Vi of (41) are small compared with u of (38), except near 

 the ends of the stream. This follows from the mode in which the exponential 

 unctions enter into (41). 



