Orr — stability or Instability of Motions of a Viscous Liquid. 75 



a system possessing an infinite number of coordinates is stable for an arbitrary 

 disturbance, however small, from the stability, even when of an exponential 

 character, of the fundamental ones into which it can be resolved ; an infinite ■ 

 series of the type 



^e~Pr^{Gr cos U)rt + Sr siu Myt), 



like one in which no exponential factor occurs, may at some times have a 

 value which is exceedingly great compared with its initial one, and may even 

 become infinite. To discover how far the motion is stable for any particular 

 disturbance, it may be necessary to obtain completely the corresponding solu- 

 tion, whether by Fourier analysis or otherwise. Possibly, it rarely happens 

 that stability for the fundamental disturbances is associated with instability 

 for those of a more general type : but this is the case in the problem under 

 discussion, as far at least as practical stability is concerned ;* this is sufficiently 

 evident from the results of Part I., and Chap. L, Arts. 11, 12, below. It would 

 seem improbable that any sharp criterion for stability of fluid motion will 

 ever be arrived at mathematically. Indeed, in simpler cases of steady motion 

 where there are only a few coordinates, although such a criterion has been 

 laid down, it has been shown that it cannot always be relied on. It has been 

 proved by Kleinf and by BromwichJ that where there is exponential insta- 

 bility, but only slight, there may be practical stability, and vice versa. There 

 is, however, this difference between such cases and the present one, that in 

 them recourse has to be had to the terms of the second order, while here the 

 motion is unstable, if terms of the first order only are taken into account. 



Chapter III., pp. 122-138, consists of some applications of the method of 

 Osborne Eeynolds. 



The method is explained in Art. 28, p. 122. Taking an arbitrary distur- 

 bance, Reynolds§ found an expression for the rate of increase of the kinetic 

 energy of the relative motion ; this is made up of two terms, of which one is 

 essentially negative, and is the dissipation function for the relative motion • 

 the other may be positive or negative. On equating the sum to zero, a value 

 of the coefficient of viscosity, fx, is obtained for which the disturbance would be 

 stationary for an instant ; if the disturbance is chosen so as to make this fi as 

 great as possible, then for any greater fx every initial disturbance must decrease ; 

 there is thus obtained an inferior limit to that vahie of /t which would permit 



* That is, if the viscosity is small enough. 



t " The Mathematical Theory ol' the Top " (Princeton Lectures, 1S96). 



X "Note on Stability of Motion with an Application to Hydrodynamics," Proc. Lond. Matli. vSoc, 

 xxxiii., Feb. 1901. 



§ " On the Dynamical Theory of Incompressible Viscous Fluids, and the Determination of the 

 Criterion," Phil. Trans. A, 186, Part I., 1895; Scientific Papers, ii. 



