76 Proceedings of the Royal Irish Academy. 



a given motion to be unstable. Previous investigators by this method have 

 selected the type of disturbance to some extent arbitrarily. 



In Art. 29, ^. 124, however, the method of variation is used to assist in 

 discovering the proper type; it is shown that when the value of jx is the 

 greatest for which it is possible that a disturbance should remain stationary, 

 the velocity components in the disturbance satisfy certain differential 

 equations. 



These are applied in Art. 30, p. 124, to the uniformly-shearing stream for 

 a two-dimensioned disturbance, supposed of definite but undetermined wave- 

 length in the direction of flow. The differential equation to be solved in all 

 such cases is linear and of the fourth order ; in this particular instance it has 

 constant coefficients. The boundary-conditions lead to equations determining 

 /z; as in the other cases to be discussed, jx, so determined, has an infinite 

 number of values ; the greatest of these is taken ; finally, the wave-length in 

 the direction of flow is so chosen that this value shall be the greatest possible. 

 The final result is BpD~liLi = 177, where p is the density, D the distance 

 between the planes, and the steady velocity is IT = By. H. A. Lorentz, who 

 discussed a species of elliptic whirls, obtained the number 288 instead.* 



Two cases of other boundary-conditions are discussed in Art. 31, p. 129. 



Art. 32, p. 130, takes up the case of a stream flowing between Jlxed 

 parallel planes, the second of the two problems discussed in such a difi'erent 

 manner by Lord Kelvin, and the numerical investigations by Eeynolds 

 himself and by Sharpe are briefly described. 



In Art. 33, p. 131, the more general plan which I have indicated of 

 using Eeynolds' method is applied to this case, again in two dimensions. 

 When the velocity perpendicular to the boundaries is expanded in powers 

 of the distance from the central plane, the differential equation gives a 

 linear relation among the coefficients of three successive terms ; there are 

 two independent solutions in series containing only odd powers, and two 

 in series containing only even ; reasons are given justifying the choice of 

 the latter (I confess I shrank from the labour of tlie double investigation). 

 The equation which determines /j. when developed from the boundary- 

 conditions is easily solved with sufficient accuracy. Choosing the wave- 

 length in the direction of flow so as to make this value of /x as great as 

 possible, there results the criterion DUpljx =117, Z7 being the mean velocity. 

 Reynolds obtained the number 517, Sharpe 167. 



Art. 34, p. 134, goes on to the case of a circular pipe, and refers to 

 Sharpe's investigation. 



* See p. 124. 



