Oer — StaMlity or Instability of Motions of a Viscous Liquid. 113 



I have not succeeded in obtaining a rigorous proof for complex values 

 of 2^- Whenever such roots occur, the approximate value, however, of the 

 real part of the first complex value of j^', as given by (57), is much greater 

 than vn^l4:a~. In fact, if a be regarded as fixed, and I is increased from 

 zero, when the first root of the period-equation reaches C, Ui being then 

 the lowest root of the equation 



J-j{32|3/aV(27v/3 v)}4 = 0, 



(which is a little greater than 7tt/12), the numerical value of p' is slightly 

 greater than (147/128) (y7rV4«^). No complex root occurs, however, until 

 / is further increased to such a value that 



Jh[32(5lay{27y3v)]i = 0, 



as the lowest value for which J^(x) vanishes slightly exceeds ll7r/12, the cor- 

 responding value of j^/ is a little greater than (363/128) (i;7r74a-). And, in the 

 approximate formula (57) for the complex roots, /, and therefore also v/^/3^ 

 has a larger value than in this critical case, while the coefficient of (v/^/3')4 in 

 the real portion is decreased in the ratio (9/11)^ ; the approximate value of 

 the real part of ^j' is thus numerically greater than 



363 /_9_\f VTT 

 128'\lTJ "4^' 



It does not seem possible that this approximate value could be so far 

 wrong that the actual value should be so small as v7r~/4:a^. 



For small values of la a further approximation to the r''' root of the period 

 equation is given by 



^ rrr (^ _ / 2 _ _10_\ ^% 



H/«/,)i._Jl_^___JC_|. (61) 



It thus seems probable that, as la is gradually increased from zero, the 

 lowest value of - j/ continually increases, and the other values of - jj' (but not 

 necessarily those of - ^j) continually decrease until they become complex. 



Aet. 23. Equations fur resolving an Arbitrary Disturhance into the 

 Fundarifiental ones : Inahility to use them. 



The problem of resolving any arbitrary disturbance (subject to the 

 boundary-conditions V~v = 0) evidently reduces to that of expressing an 

 arbitrary function of y which ^'anishes when y = ± a, in terms of the 

 functions 8 which correspond to the free modes of disturbance already 



