1892.] and Stability of Dynamical Systems. 355 



where 0, <p, e,f are small angles. In the first case, 5? is a minimum 

 in steady motion. In the second case, in which rotation takes 

 place about the greatest axis, £ is a maximum ; but it can be 

 shown by other methods, that the steady motion is stable. In the 

 last case, in which the rotation takes place about the mean axis, 

 £ is a maximum for some disturbances, and a minimum for others; 

 and the motion is well known to be unstable. 



The conditions of stability, when there are two coordinates 

 of the type 0, are given by Routh*, and are somewhat com- 

 plicated. The general conditions of stability, when £■ 4- V is 

 not a minimum in steady motion, do not appear to have been 

 investigated. 



7. The advantages of the Theory of the Ignoration of Co- 

 ordinates is most strikingly illustrated in Hydrodynamical pro- 

 blems; for in this subject, the most convenient form of the 

 kinetic energy is frequently one, which is not entirely composed 

 of velocities, which are the time variations of coordinates. More- 

 over, the generalized velocities corresponding to such quantities 

 as components of molecular rotation, vorticity and the like, are 

 frequently unknown, or would be troublesome to introduce. We 

 shall presently call attention to two Hydrodynamical problems, 

 in which the power of this method is strongly brought out. 



It must however be noticed, that it is not necessary that 

 the quantities k should be momenta in the ordinary sense of 

 the word; for if r t , t 2 ... be any other quantities, which are con- 

 nected with k v k 2 ... by a series of relations of the form 



K i=fA T i> T 2> ••• )■ 



equations (2) and (3) would still apply ; provided the functions 

 f do not contain any of the coordinates 0. This remark is of 

 some importance, as a convenient transformation often shortens 

 the work. 



8. In the case of a cylinder moving parallel to a fixed wall, 

 when there is circulation f, the kinetic energy is 



T = 1R (x 2 + f) + K 2 paj4!7r, 



where k is the circulation. 



The coordinate x is an ignored coordinate, and the constant 

 momentum h, corresponding to x, is the momentum of the system 

 parallel to the wall, which is equal to 



h = Rx + icpc ; 



* Stability of Motion, p. 88. f Hydrodynamics, vol. i. § 213. 



