) 



Orr — Stahility or Instability of Motions of a Viscous Liquid. 129 



Akt. 31. Tioo instances of other Boundary -Conditions. 



As another example, suppose the former boundary-conditions are replaced 

 by 'y = 0, d^vldif = 0, equivalent to -^ = 0, drtpldy- = 0. Equation (13) has 

 now to be replaced by 



(lUimz + mz-m-i-) sin [nii - m2)a sin (ms - m^) a + {ii^^m^ + m-^hiii) sin (77^2 - wis) « 



sin (?/ii - m^a + {niziiii^ + 171^111^) sin {m^ - nii) a sin (nii - mi) a = 0, (39) 



or, in the notation of (14), 



{(r- - r'^)^- 42r(7'^ + r'^) jsin2?'asin2/a + 8pV?^'cos2racos2r'<:t - 8_^9V7''cos4pa = 0. 



On writing again r = iq, r = iq, this becomes 



{(2^ - (f^-y + 4/:'X2^ + ^'^)}sinh 2qa sinh2g''(X + Sp'^g'/cosh 2g'a cosh 2g^a 



- ^p^qq'coQ 4ipa = 0. - (41) 



As the first two terms are positive, and the second exceeds the third 

 numerically, this equation cannot be satisfied, and, accordingly, as before, 

 we fall back on the other alternative, viz., r real and r imaginary. Writing 

 in (40) r' = iq' simply, it becomes 



[{r- + q"y' + Af^if^ - r*)J sinh 'leqa sin Ira + '^]f^r cosh Iqa cos 2ra 



- '^jfqr cos Aq)a = 0. (42) 



Now this equation has no solution for which 2ra is less than 7r/2 ; for within 

 this limit, as q"^ > 3r^, the left-hand member is certainly algebraically greater 



than 



8p~r {r sinh 2q^a sin 2ra + rf cosh 2qa cos 2ra - q cos 4:pa] ; (43) 



and while 2ra- increases from to 7r/2, the sum of the first and second terms 

 in the brackets increases continually, and therefore everywhere exceeds its 

 initial value q; hence the result follows. We may, therefore, equate 

 sinh 2q'a and cosh 2q'a, and neglect cos 4jj« in comparison. Thus we ha^'e, 

 expressing the coefficients in terms of p, I, 



tan 2ra = - i {?>f - I'f {if + rfK (44) 



and the lowest value of 2ra accordingly lies between Stt/G and w. As a 

 condition for a stationary value of /x, we now obtain, using (32), 



ap (Sp^ + 21-) (32^ - I'f dridp = 3/- {p^ + l~)K (45) 



and, by the aid of (21), (32), (34), there results, instead of (36), the equation 



2ar[^p-' + 21') {2p (jf + P)i - 2^ - 2P) = Qp^P (2p - (jf + l')i) (3/ - l')-k (46) 



