100 Proceedings of the Royal Irish Academy. ~ 



'\'\'Tiile, if the boimdary-conditions were that E should vanish at one 

 boimding-plane, and dSjdij at the other, it may be seen that the period- 

 equation has an infinity of roots, and that all the values of y have negative 

 real parts numerically greater than v\- ; the conclusion that the imaginary 

 part of 23 lies between the limits ± li^aA would not, however, hold. And in 

 the right-hand member of (14), m would be replaced by {2r + l)7r/2, as we 

 should now require, approximately, Ae''- + Ber^' to vanish for one value of the 

 parameter, and Ae^ - Be~^ for another, so that the two values of the parameter 

 would differ approximately by [2r + V)nil2. 



It thus appears that the fundamental modes of free disturbance possess 

 stability of the ordinary simple exponential character, when the boundary- 

 conditions include the vanishing of V'v. 



Aet. 16. For all values of I, n, there are o/ii infinite numher of A'periodic 



Dishcrhances. 



Considering real values of jj for which v\- + p is negative, if we take that 



value of 3 



vX" + p .) ^ 



whose argument is zero when y is zero, then when y is a, its argument must 

 lie between the limits and 37r/4 ;* and when y is - «, its argument must lie 

 between and - 37r/4. ISTow, from (9), (10), there is one linear function of 

 I^kif'^) and Ik{r), viz., a multiple of /-^ (•:':) - Ik{x), which, for large values of x 

 whose argument lies between - 877/2 and + 37r/2, is approximately x~ie~^ ; 

 and there is another function, viz., a multiple of I-k{x) + Ik{x), which, when 

 the argument lies between and tt, assumes the approximate form 



x's {e^ + i cos lin . e"^), 



but wliich, when the argument lies between and - tt, is approximately 



x'i [e^ - i cos k-n . e'^). 



If, then, we write 



2i/8/ vX' + p .V(* 2i7/3/ vX- + 7J A 



«0S^ = ^^- ^F(-^^^^7^-'^0?=«^^ (22) 



O \ V \ 



the period equation is, approximately, 



e"i + ij'l . e-"i _^ e"2 - i/2.e-"2 



* This is true for complex values also, since, as proved in Art. 15, the imaginary part of j/A.'- + p 

 lies between the limits ± l^ai. 



