Orr — Stability or Instahilitij of Motions of a Viscous Liquid. 103 



the approximate expression for it in the different portions of the region 

 traversed. 



In fig. 1, denoting the origin, let A, A' on the axis of imaginary 

 quantities denote the points ^lai, - (5lai ; through A draw AL parallel 



L'- 



FlG. 1. 



to the axis of real quantities and in the negative direction, and draw AM, AN 

 making angles of 27r/3 with AL ; also draw A'L, A'M\ A'N' parallel to 

 AL, AM, AN. Suppose 2^' starts from a point on the line AL; let the 

 argument of each power of Ui be zero in that position ; and let the argument 

 of each power of 11^ be zero when p' moves down to AL'. When p' lies 

 between AL, AL' , since the ratio of its value, given by (28), to ^la is 

 large, the argument of u^ is a small positive quantity, and that of Uz a 

 small negative quantity. Thus, in this region, from equations (9), (10), (11) 

 we have 



u^^{L.{u,) - ^{tH)) = (2/7r)J sin 7r/3 . 6-1, (30) 



u^il-liu;) + /. (t^O) = (2/Tr)* (e"i + il2 . e-^), (31) 



u,i(L,(u,) - L,(u,)) = (2/7r)^sin 7r/3 . 6-"^ (32) 



u,i(L.Xu,) + Ii(u,)) = (2/7r)Ke-^ - ^/2 . e-2) ; (33) 



so that, omitting a constant factor, the left-hand member of (29) has the 

 approximate form 



6-«i (g"2 - i/2 . e'"-i) - e-^'-i (e"i + i/2 . e""i ), (34) 



gu.,-uy _ g!<i-«2 - ie-^h-'^i, (35) 



or. 



When p crosses to the lower side of AL', since the argument of ih then 



[14='=] 



