()rr — Sfabiliti/ or InstabiUtij of Motions of a Viscous Liquid. 93 



Art. 12. A similar Inmstigation for the Second Modification. 



If we take the solution which would make dvjdy, and therefore u, zero, 

 instead of v, at the bounding planes, it is seen that the two- dimensioned form 

 corresponding to an initial disturbance in which 



V = v^ = B sin Ix cos my (44) 



has a stream function given by 



■^l^p sinh lb Exii [- vt(f + m- - Im^t + /-/3^f/3)] 

 (/■- + mr)B " t' + {m - l^tf 



X I sinh lb cos \lx + (m - l^t) y'] - ^' {m - l^t) cosh l{b - y) sin Ix 



+ Z-^(m - l^t) cosh Z?/ sin \lx + (m - l(St) &] } 



+ another term derivable by changing the sign of m. (45) 



In this case, the ratio of increase at time t is 



T P + m' 



T 2 



' Exp[-2vt{P+m''-lm^t^l'p;^fl2,)'\ Ux2j[-2vt(l'+m'-vlm.l5t^l-[iHy3)] 



+ 



l'+(m-l(5ty l'+(m+lldt) 



+ ' 



{m-l(5t)JExp[-vt{r'+m-'-tm[5t+P(5Hy3)]Jm+l(5t]mpl-vt{lHm'+lm(5t+l'j5Hy3^^ j 2 



cosh lb - cos (m - l(5t)bl 



^Ib sinh /5 _ ^ ' 



Here, again, there is a possibility of a large increase if m,/l is large.* At the 



instant when m - Ijit is zero, the only term in this which is not negligible 



assumes the form 



^ (-2 vm ... ,, ^, I P + m- , . ^^ 



simpler than (37), and capable of assuming a much greater value. A condi- 

 tion that (47) should be a maximum is 



V {P + m'Y = hn(3, 



or vm.' = I ft ; (48) 



and then it is approximately 



6-tj3V2mS'^ or e-i¥(3\-y2m'b\ (49) 



If ftb^/v = 1900 and mb has its lowest value, tt, this is nearly 9500. Taking 

 ftb'/v = 1940, we have in round numbers 10000. 



*- It is not evident that, as in the case of the first modification, there is no possibility of a great 

 increase under any other circumstances. 



K. I. A. PROO., VOL. XXVII., SECT. A, I •'■^] 



