Motion of a Viscous Incompressible Fluid. 335 



as appears from the value assumed when 77 = 0. Thus 



k-i% <»> 



which defines S 2 in terms of Si. 



A similar relation holds for any two particular solutions. 

 For example, 



h"£% <»> 



The difficulty of the stability problem lies in the treatment 

 of the boundary condition 



P 8, e X "d v . ("'■ S, e-^dr,- f" 2 8, <T X % . ( "' S s /•*=<> 

 Jii J'/i Ji. Jii 



. . . (31)« 



in which tj 2 , rj u and X are arbitrary, except that we may 

 suppose r] 2 and X to be positive, and tj 1 negative. In (31) we 

 may replace <? x?? , e~ Xr} by cosh Xrj, sinh X77 respectively, and 

 the substitution is especially useful when the limits of 

 integration are such that r) i = — r) 2 . For in this case 



1 S cosh Xy drj = 2 I s cosh Xrj drj, 

 Jli Jo 



I S sinh \r) drj = 2i I £ sinh \i? (£17 ; 



Jn, Jo 



and the equation reduces to 



I 2 Sj cosh Xrj drj . \ " t 2 sinh Xrj drj 

 Jo Jo 



— I 2 s 2 cosh X77 <^?7 . I 2 ti sinh Xt; cfy = 0, (32) 



Jo Jo 



•thus assuming a real form, derived, however, from the 

 imaginary term in (31). In general with separation of real 



* Rather to my surprise I find this condition already laid down in 

 private papers of Jan. 1893. 



