Orr — Stability or Instability of Motions of a Viscous Liquid. 121 

 If \a is made to diminish without limit * this becomes 



S,dy 



S,ydy- S^ydy S,dy = 0. (76) 



111 the region in which the roots actually lie. this assumes the approximate form 



- {(6"i -'ie-"i)%i^ - e"2%2^} [e-''^ur' - e-'-^nc^'^} = 0. (77) 

 For real roots, if p' is not too near C (fig. 2) , this may be replaced by 



identical with (14). Even in this somewhat simple case, the equation giving the 

 complex roots does not appear readily solvable. In this case it may be shown 

 that the critical point at which p' becomes imaginary does not coincide with 

 C (fig. 2) ; but that some of the roots become imaginary at points to the left 

 of C, and others at points to the right ; that for the roots which are of low 

 order the absolute distance of the critical point from G is not large, and that 

 as the order of the root rises it tends asymptotically to C. The complex roots 

 thus consist of four series — one to the left of AC, another to the right, together 

 with the images of these series in the axis of real quantities. 



In the most general case the critical point at which roots become imaginary 

 is not far from C; and the values of ^' lie not far from the lines AG, A'G. 



It is thus seen that, unless either \a is large, or else [ila^/v so small that 

 all the disturbances are aperiodic, the results I have indicated are very 

 incomplete for the natural boundary-conditions v= 0, dv/dy = 0. 



* If the velocity-gradient is great enough, \a may be very small, and yet fila^/f not small ; so that 

 for suflaciently rapid motion this case is a little more general than that in -which v is made a function 

 of y only. In the latter case, the method similar to that of Art. 16 succeeds in proving directly that 

 the disturbances are exponentially stable ; this result was, I believe, obtained many years ago by Love, 



