Orr — Stability or Instability of Motions of a Viscous Liquid. 117 



where r is zero or any positive integer ; and, more accurately, by 



Jlf I %i I - 7|| u, 1 = 0; 



the second term of (66) is now small compared with the other two. These 

 complex values of p\ of course, as before, lie close to the line CA, and their 

 conjugates close to CA\ 



It is seen that here again all the roots which exist have been accounted 

 for and approximately located. 



It will be noticed that, approximately, when la is large, the real roots, if 

 not too near C, are the same as when the boundary-conditions are >S' = ; the 

 complex roots are different, however ; this is the only evidence I have noticed 

 against the view that, for disturbances whose wave-lengths in all directions 

 are small, the question of stability is little affected by the precise boundary- 

 conditions. 



Aet. 25. The Case of Boundary-Conditions V=0, dV/dy = 0: Failure to obtain 

 any Simple Proof that fmdamental Disttirhances are Stable. 



With the boundary-conditions V= 0, d F/dy = 0, I am unable to give any 

 simple proof by any method analogous to that of Art. 15 that the funda- 

 mental modes of disturbance are exponentially stable. We obtain, however, 

 the same limits for the imaginary parts of the values ot jp, viz., ±l^ai. The 

 equation satisfied by V being 



[d'Idf -[\'+ (p + il[iy)lv}]{dydf -X')V=0, 



if we write V=Vi + iVi, ^j = + i(p, separate the real and the imaginary parts, 

 multiply one equation by Fi, the other by Fa, add, and integrate between the 

 limits ± a, we readily obtain 



(r/. + Ifty) [(d V.ldyf + {d rSvY + A'- ( Fr + V^)]dy = 0, (69) 



from which it follows that ^ + l[5y must change sign between the limits of y. 

 I have also been unable to obtain any equations analogous to (63), (64) 

 Art. 23, by the aid of which any arbitrary free disturbance may be resolved 

 into its constituent fundamental ones. 



Aet. 26. Derivation of the Period- Equation : Its approximate Form. 

 The solution of (1) being denoted by >S', V may be expressed in the form 



Se-^ydy - e-^y Se^ydy\ , 

 J / 



V-^^U^' 



whence dVjdy = \\e^y 



Se'^ydy + e' 



Se^ydyX 



E.I. A. PROC, SECT. XXVII., SECT. A. [16] 



