Orr — Stability or Instability of Motions of a Viscous Liquid. 125 



Analogy with other problems leads us to assume that disturbances in two 

 dimensions will be less stable than those in three ; this view is confirmed by 

 the corresponding result in case viscosity is neglected, seen by comparing 

 equations (28), (38) of Part I., Chap. I. ; it is further strengthened by com- 

 paring the two- and the three-dimensioned forms of equation (29), Chap. I., 

 above, and by the discussion of the fundamental free disturbances in Chap. II. 

 Considering, then, the two-dimensioned case,* the elimination of ^ from (7) 



gives 



2fxV^{cUildy - dv/dx) - pB {dvjdy - du/dx) = 0. (8) 



We may now conveniently introduce the stream-function \p, when this becomes 



fxW'xP + pBd^/dxdy = 0. (9) 



This is to be solved subject to the conditions that xp and dxp/dij vanish at 

 the bounding planes which we will denote by ^Z = ± ct. We next suppose that, 

 as a function of x, \p varies as e*^-^, when the equation becomes 



fiidyd'if - iy4> + ilpBd^p/dy = 0. (10) 



The fundamental solutions are xp = e""^/ where the values of m are given by 



n(m^ + I'f - Bphn = 0. (11) 



Denoting the roots of this by mj, mj, ma, m^, the equation to which the 

 boundary conditions lead is 



7«, ai 





m„ at 



tn^ni 









= 0, (12) 



or 



{nfhim^ + m^mi) sin (nii - m^) a sin {niz - m^ a 



+ {friz'nis + miVi^ sin {rti^ - tn^ a sin (wij - m^ a 



+ {m^nii + mzfrii) sin (ms - mi) a sin (mo - m^) « = 0. (13) 



As the sum of the values of m is zero, they may be written 



p + r, p - r, - p + r\ - p - r\ (14) 



where p is real, and, making these substitutions, (13) becomes 



"(4^^ - r"- - /2) sin 2ra sin It' a - 2r/cos 2ra cos 2r'ci + 2?'r'cos A^pa = 0. (15) 



Now, the values of m which satisfy (11) must all be imaginary, or else two 

 real and two imaginary. 



*■ The ttree-dimensioned ease was attempted, but it proved too difficult. 

 "- — — ■ . [17] 



1. 1. A. PROC, VOL. XXVII., SECT. A. 



