110 Proceedings of the Royal Irish Academy. 



Now, the roots of the equations </_i(.k) = 0, Jl{x) = 0, occur alternately; 

 those of the former are approximately cc = rn- + lirjl'l, and those of the 

 latter x = rir + ll7r/12, where r is zero, or any positive integer; and, as has 

 been proved in Art. 19, whenever the value of j^' at C is such that the 

 corresponding value of v.ii(oT iui) is a zero of J^icc), this value of p' is 

 a double root of the period-equation. Hence we can trace the effect of 

 dimuiishing the wave-length in the direction of flow on the nature of the 

 roots of the period-equation. Starting with a very small value of la, if we 

 gradually increase it until 



||^(^«J|' or (32//3«V(27v/3..))i (51) 



becomes equal in value to the lowest zero of J-l{x), the smallest value of 

 y is represented by the point C ] if we further increase /, this value passes 

 between C and 0, and so remains until the expression (51) becomes equal 

 to the lowest zero of ^|(») ; at this stage two roots of the period- equation 

 coincide at C. On increasing the la still further, these two roots become 

 complex, and there is now no root between C and until the expression (51) 

 becomes equal to the next zero of J-\{pi), at which stage a root passes C, to 

 return to it, and, coalescing with another, become a double one when (51) 

 becomes equal to the second zero of J^ (.'•) ; after this these two become 

 complex and different ; and so on. 



That a pair of roots do, indeed, become imaginary as la increases through 

 the value which makes them coincident, may be seen as follows : — It has been 

 shown that when ht is sufficiently small, there is one, and only one, root 

 between the real values for which 



u, - u^ = (2r ± l)TTil'2 ; (52) 



now, the roots are continuous functions of a, i.e. d{p'lcla is finite (except when 

 jp' is a double root) ; hence, the only manner in which this distribution of 

 roots could be altered would be by a root passing through a point given 

 by (52). But, by making use of the above expressions for the limits of error, 

 it is easy to prove that this is impossible ; thus, two real roots do disappear — 

 one from the left and one from the right of C — while the value of Ui at C 

 changes from (7* - \) -wi to {r + \) tri. But, from the statement in the final 

 sentence of Art. 18, p. 105, these roots continue to exist, and must therefore 

 be complex. 



Thus, the greatest wave-length in the direction of flow for which a 

 disturbance can be oscillatory is 27r//, where 



{32/3/aV(27v/3).)l^ = the lowest zero of Ji(.r) = 2-87. (53) 



