Ork — Stahility or Instahility of Motions of a Viscous Liquid. 115 



doing, the evaluation, accurate or approximate, of the coefficients A by means 

 of (65) would be only one step. Were this accomplished, we would have 



S = i ArSr{y)er'r\ (65 a) 



and V would have to be found from this, by the aid of (2), and found in a 

 form suitable for arithmetical comparisons. 



It may be noted that although, from the results of Chap. I., above, and 

 those of Part I., there is good reason to suppose that, for a suitably chosen 

 initial disturbance, F may increase very much, this is not the case with S. 

 On the contrary, it readily follows from (2) of Chap I. that the average 

 value of /S'2 throughout the liquid diminishes continuously and indefinitely ; 

 a similar contrast between decreasing >S' and increasing V may be noted for 

 the disturbances discussed in Chap. I., Arts. 2 and 10-12. 



Aet. 24. The Case of Boundary-Conditions dSjdy = 0. 



If the assigned boundary-conditions are that dSldy should vanish at each 



of the boundary-planes, the period-equation is obtained by making, in the 



notation of equations (5), (6), (7), 



^'f(F)+^V(F) 

 vanish at the boundaries ; but 



^'(F) = 3-%(-f)F/_|(fF'x 



2f (F) = 3%(|)F/|(fF'); 

 so that the equation is similar to (8), except that the / functions are of 

 order ± -|. 



For large values of p' whose real part is negative, the approximate form of 

 this equation is 



gWi-«.2 _ gM2-Mi _ ^'g-«i-M2 ^0. (66) 



Obviously it may be proved, as in Art. 16, that for all values of /, n, there 

 are an infinite number of aperiodic disturbances, the values of ]j being given 

 approximately by (14), (15) again. 



Evidently, too, if ta is small enough, in (15) r may be taken to be any 

 integer, even unity. 



But an investigation almost identical with that of Art. 18 proves that, for 

 all integral values of r (including zero), if la be small enough, and for all 

 values of la, if t be large enough, the number of roots inside the circular 

 contour for which 



mod 2a[-p'lv)\= {r + Dir 



is r + 1, one more than vnth the houndary-conditions S = 0. This difference in 



