Orr — Stahilitu or Instabilitjj of Motions of a Viscous Liquid. 99 



Art. 15. Each Fundamental Disturhance satisfying the Boundary -Conditions 



V~y = is exponentially stahle. 



It may next be shown that all the values of 'p which satisfy the period- 

 equation (8) have a real negative part. This follows easily by a method 

 which has been used by Lord Eayleigh in the discussion of similar questions 

 when viscosity is ignored. The period-equation has been obtained by making 

 the function 8, which is a solution of equation (1), vanish for the two values 

 y = ± a. In equation (1), then, write S = P + iQ, 'p = 9 + i(p, where F, Q, 0, (p 

 are all real ; separating the real and imaginary parts we have 



vd^F/chf = (vX'- + Q)F - (^ + liiy) Q, (16) 



vcf'QIdf = (vX' + 6) Q + (({> + l^y)F. (17) 



Multiplying the former by F, the latter by Q, and adding, we obtain 



v{Fd'Fldf + Qd'Qldy'') = {yV + %) {F~ + Q"). (18) 



Integrating with respect to y from y ^ - a to y = + a, since S, and therefore 

 both F and Q, vanish at the limits, we obtain 



vj_^ {[dFldyf ^- {dQldyr-]dy 



{vX' + 0)(P^ + Q')dy, (19) 



The right-hand member must therefore be negative, so that not only must p 

 have a negative real part, but that real part must be numerically greater 

 than vX^. 



If we multiply (17) by F, (16) by Q, and subtract, we obtain 



viFcl^Qldf - QcPFIdf) = (v> + //By) (i^ + Q'). (20) 



Integrating with respect to y from y = - a to y = + a, since F and Q both 

 vanish at the limits, we obtain 







(0 ^iMiP'^Q'y^y^ (21) 



so that ^ + /j3.?/ must change sign as y passes through some value between 

 - a and + a. Accordingly the value of <^ must lie between the limits 

 + /|3a. 



If the boundary-conditions assigned were that dSjdy should vanish at the 

 bounding-planes, it may be readily seen that all the conclusions drawn above 

 as to the existence of, and the nature of, the roots of the period- equation still 

 hold. 



