Orr — Stability or Instahility of Motions of a Perfect Liquid. 29 



which initially vanish as the original value of ic satisfies the end-conditions. 

 Find Cr, Dr, other functions of t, such that 



. - ^ (40) 



C^ er'"^lb + Br e-''"'''^ = Br 



Take then 



^h = - 2 { Gr e^'^^/^ + Dr e-'-'^^/'^) cos rwtjlh, 



' (41) 



Vi= 2 {Or eT'^l'' - Br 6-'-'^^/*} sin rirylh 



r=0 



these are the values to be added to %i, v, as given by (38), in order to complete 

 the solution. 



In questions similar to that now under discussion, the use of infinite series, 

 such as occur in (39), sometimes requires justification, especially in regard to 

 differentiation. On such points reference may be made to Stokes' classical 

 memoir.* In the present case, as the series in (39) converge, the form in which 

 the exponential functions occur in the series in (41) shows that these latter 

 series, as well as those formed of the differential coefficients of their successive 

 terms with respect to x or y, are uniformly convergent in the space considered ; 

 and in this space the differential coefficient of any order with respect to x or 

 y of the sum of the series is evidently accordingly the sum of the differential 

 coefficients of the separate terms; and thus the complete values of Ui, v^, as 

 well as the separate terms, satisfy the differential equations. The vanishing 

 of the series for Vi when y = or h, or rather when y is just inside these 

 limits, does not follow from the mere fact that it is of the form 2«,- sin riry/h, 

 but is secured by the additional circumstance of its uniform convergence at 

 these limits. 



Art. 7. Another Examjjle of Prescribed End-conditions. 



As another example, if the prescribed end-conditions require u to vanish 

 at two planes x - ^ty = {), x- (Sty = a, which move with the fluid, we replace 

 the quantities Mi, Vi just found, by others obtained in a similar manner from 

 the values of to of (38) at these planes instead of from its values at the fixed 

 planes. 



Aet. 8. The Solution foimd explains Instahility. 



If the conditions to be satisfied at the ends of the stream are either 

 those of (6) or those of (7) (and the same holds for many other conditions 



* " On the Critical Values of the Sums of Periodic Series," Camh. Phil. Trans., viii., p. 533 ; 

 Collected Papers, t. i. 



