72 Proceedings of the Uoyal Irish Academy. 



y = ± a, and the steady velocity to be j3y in the ^.'-direction, the ^/-velocity 

 in the disturbed motion is taken to be v = Ye^'t+i{i^+n~)^ where I and n are 

 arbitrarily assigned and ]3 is to be found. The differential equation shows 

 that V^v is of the form : — 



u'^[AJx_{u) + jBJ_i{u)} where u is of the form (Cy + C")l ; 



if the boundary-conditions should include the vanishing of V^v, it is thus seen 

 that the investigation is very much simpler than for the natural conditions 

 V = 0, dv/di/ = ; and accordingly this case is discussed in detail. 



In Art. 13, p. 95, the equation giving the values of |) (the period-equation) 

 is derived. 



In Art. 14, p. 96, in view of a remark of Lord Eayleigh's which appears 

 to suggest that it may not be possible to obtain disturbances which do vary 

 as d^\ it is first proved, or rather rendered probable — for the demonstration is 

 not rigorous — that this equation has an infinite number of roots ; this follows 

 by making use of the approximate forms of the Bessel functions for large 

 values of the variable. 



In Art. 15, p. 99, it is proved directly from the differential equation that 

 all possible values of p must have a real negative part, and that the imaginary 

 part lies between the extreme limits found when there is no viscosity. 



Art. 16, p. 100, gives a rigorous proof that for all values of I, n, there are 

 an infinite number of real values of ]). 



Art. 17, p. 101, indicates briefly a proof that if la is small enough, all the 

 values of ]) are real, and given approximately by a comparatively simple 

 algebraic equation ; this proof is developed rigorously in Art. 18, p. 102, which 

 contains as a necessary step an investigation of the number of roots inside a 

 circular contour of large radius having the origin as centre, this investigation 

 and its result holding good, whatever the value of la. 



In Art. 19, p. 106, the double roots are considered; it is shown that a 

 double root occurs when, and only when, a certain multiple of (l(5a^/v)^ is a 

 root of Jiiii-) = 0, V denoting the kinematic viscosity; and, in Art. 20, p. 108, 

 it is proved that, as / increases through such a value, two real roots do actually 

 disappear; while in Art. 21, p. Ill, approximate expressions are obtained for 

 the complex roots. It is seen that all the roots, real and complex, are 

 accounted for. There are thus a definite finite number of complex roots, and 

 for them the values of j^5 + v (P + n^) lie close to two straight lines which 

 contain an angle of 27r/3. When the disturbance is oscillatory, its time is 

 independent of n. 



In Art. 22, p. Ill, it is proved that, in the most persistent disturbance, v 

 is a function of y only ; i.e., I and n are zero. 



