74 Proceedings of the Roi/al Irish Academy. 



In Art. 26, the period equation is expressed in terms of integrals which 

 involve V^v, a function whose form has heen already found. On the 

 supposition that the approximate forms of the Bessel functions, for large 

 values, may be used in this case also, I have given an approximate form 

 of the equation appropriate to the region in which the roots actually lie. 

 In this portion of the investigation somewhat intricate questions arose from 

 the fact that the approximations assume different forms in different regions. 

 Fortimately, in the region in which the roots actually occur, the difficulty 

 is not met with in its entirety. As I am quite unable to solve this 

 equation in the most general case, it seems undesii'able to give this portion 

 of the investigation, which is somewhat long, in full. 



In Art. 27, p. 119, some results are stated. It appears that for none 

 of the roots can the disturbance be unstable, but owing to the way in 

 which approximations have been used, the proof indicated is not rigorous. 

 The result of an investigation of the number of roots inside a circle of 

 large radius round the origin is stated. The period-equation for a liquid 

 at rest, a problem discussed by Lord Piayleigh, is obtained as a special case. 

 A reference is made to the case in which a (I- + n'^)^ is large ; for the 

 smaller values of j^ the roots are very nearly the same as with the boundary 

 conditions V^v = 0. Some reference is made to the general case ; for such 

 of the real roots as are remote from the complex ones, an equation is given, 

 which, if the values of the constants were given, could be readily solved ; 

 for the others, especially the complex ones, the form is very complicated. 

 In all cases, however, there are an infinity of real roots, and a finite, but 

 undetermined nimiber, which may be zero, of complex ; and, roughly speaking, 

 for these the values of p + vQ' + n'^) lie in the neighbourhood of the same 

 two lines as with the boundary-conditions V^v = 0. An approximate form 

 of the period-equation is given suitable to the case in wMch a (l^ + n^y is 

 indefinitely small, the form of the period-equation pre%dously taken now 

 becoming an identity ; the equation giving the complex roots is still 

 complicated. 



It will be seen that, except in the case of very slow motion and in 

 that of large ^'alues of a {P + n^)^, the discussion is very incomplete and 

 unsatisfactory when the boundary -conditions are that v and dvlcly should 

 vanish. 



Owing to the failure to use Fourier analysis in the simplest case,* the 

 whole investigation elucidates the question of stability but little ; for it seems 

 unjustifiable as a mathematical proposition to infer that the steady motion of 



* I. e. that in which the houndaiy-conditions include the vanishing of V'f . 



