Dr. T. J. Fa. Bromwich on Kinetic Stability. 71 



functions of the time; although of course any single term 

 in at will be replaced by a pair of terms containing (a + co)t 

 and (a — a))t. 



What appears to have been overlooked by Mr. Jeans is 

 that these systems are not rendered unstable until frictional 

 terms are introduced, the amount of the friction being pro- 

 portional to the relative velocity ; the facts are clearly 

 brought out in two examples given by Lamb *. 



A similar oversight appears to occur in the associated 

 gyrostatic problem (of constant angular momentum) men- 

 tioned in Art. 29 : incidentally, the equations there used are 

 attributed to Schwarzschild (quoting papers dated 1896 and 

 1897). They appear to me to be simply a special example 

 of Routh's general process f for ignoration of coordinates^ 

 published in his Adams Prize Essay of 1877. 



But before leaving the topic of stability, it may be worth 

 while to refer to the fact that errors have been made by 

 attempting to infer instability from the method of small 

 oscillations. The classical instance is the top, sleeping 

 upright with such a spin as to satisfy exactly the critical 

 condition of stability ; then (using the method of small 

 oscillations), Routh deduced that this top would be really 

 unstable (Adams Prize Essay). About twenty years later 

 Klein proved in his Princeton lectures that this top is really 

 stable J. 



It does not appear to be possible to give any complete test 

 to settle the question of stability in these critical cases ; but 

 a number of general considerations are given in a paper of 

 my own on this topic §, together with a detailed examination 

 of the allied problem of a solid moving through liquid, 

 accompanied by circulation round the solid; in this problem 

 the critical cases may be either stable or unstable, according 

 to a rather elaborate additional criterion. 



It may be useful here to quote from Klein's paper, in 

 reference to the general question of trying to discuss 

 problems of stability by means of the method of small 

 oscillations : — 



"From the start this method of small oscillations lies 

 open to severe criticism. In the so-called unstable case it is 

 directly self-contradictory, since the quantities, which in the 



* Proc. Roy. Soc. (A) vol. lxxx. p. 168 (1908) ; or 'Higher Mechanics/ 

 Art. 99, exs. 2, 3. 



+ See, for instance, Lamb's ' Higher Mechanics/ Art. 83. 



% Bulletin of the American Mathematical Society, vol. iii. p. 129 

 (1897). 



§ Proc. Lond. Math. Soc. (1) vol. xxxiii. p. 331 (1902), 



