Ore — Stability or Instability of Motions of a Viscous Liquid. 91 



of T thus found with its initial value, i.e., T^ = B^l (/- + m'^)l^l'^, there results 

 T _ (/" + m-) V Exp [-2 vt (lHm'-lml5t+l'(3Hy3)'] Exp [-2vt {P+77v-^lm(5t+P(iH-/3)] 



,, cosh lb - cos (m - l(5t) b' 

 ilbsinhlb 



(34) 



As the terms to be subtracted from the two first are essentially positive, 

 there is no possibility of any great increase, unless the first two are large ; 

 and even in the absence of the exponential factors, this can occur only if 7n/l 

 is large, and then solely during the time in which m - Ifit is of order not 

 larger than /. At such times, the terms which have /" + (m + //3^)^ in the 

 denominator may be neglected in comparison with the others. During such 

 a time, if qti/I is large, we may approximately replace* the exponential factor 

 of the two remaining terms by Exp {- 2vm^/{3l(5)}, and thus obtain 



T^-^m^( P cosh/5-cos(m-/j305 j ^^^x 



To" 2P\ P+(m-l[5ty ilbsinhlb 



As the last factor is less than unity, a large value for T/T^ requires that 

 vm^lQ^) should not be large, i.e. that m^5^ . mjl should not be large compared 

 with ^b'^jv ; now the smallest possible value of mb is tt, and mjl is large, 

 so that instability requires ^b'^jv to be large. Conversely, if m//, mb are 

 large, and jS&Vi' large enough to be of the same order as m^b"^ . mjl, an initial 

 disturbance of the type given by (28), and subject to the boundary-conditions 

 required by (29), (30), one of which is v = 0, will increase very much before 

 dying out. At the time when m - l^t = 0, we have in fact 



T . zl]^ m~ ( . tanh ilb) . .^^. 



e 3/^ -^Jl iTT— • (36) 



^0 ■ 2r- ( ilb , 



With the relative magnitudes chosen for the constants, the exponential factor 

 is not small, and the product of the other factors is large, its approximate 

 values in the extreme cases of lb large and of lb small being respectively 

 my2P and wt'&V24. 



It may be of interest to take values of the constants for which a somewhat 

 similar motion has been found experimentally to be unstable, and ascertain 

 to some extent how much they would allow a disturbance of the type (29), 

 (30) to increase. Couette foundf that, when a cylinder of radius 14"6395 cm. 



* I.e. in the sense that lliis gives the index uf the exponential factor with sensible accuracy, 

 t " Etudes sur le frottement des liquides," Annales de Cliimie et de Physique, (6) xxi., p. 433, 



