230 Notices respecting New Boohs. 



A Treatise on the Stability of a given State of Motion, particularly 

 steady Motion; being the Essay to which the Adams Prize was 

 adjudged hi 1877, in the University of Cambridge. By E. J. Eotjth, 

 M.A., F.B.S., Sfc. London : Macmillan and Co., 1877. (8vo. 

 Pp. 108). 



The question, to which this Essay is an answer, was proposed in 

 the following words : — " The Examiners give notice that the fol- 

 lowing is the subject of the Prize to be adjudged in 1877 : The 

 Criterion of Dynamical Stability. To illustrate the meaning of the 

 question, imagine a particle to slide down inside a smooth inclined 

 cylinder along the lowest generating line, or to slide down outside 

 along the highest generating line. In the former case a slight 

 derangement of the motion would merely cause the particle to 

 oscillate about the generating line, while in the latter case the 

 particle would depart from the generating line altogether. The 

 motion in the former case would be, in the sense of the question, 

 stable, in the latter unstable. The criterion of the stability of the 

 equilibrium of a system is, that its potential energy should be a 

 minimum ; what is desired is a corresponding condition enabling us 

 to decide when a dynamically possible motion of a system is such, 

 that if slightly deranged, the motion shall continue to be only 

 slightly departed from." 



In very brief outline Mr. Eouth's answer to the question is as 

 follows : — When a dynamical system is making small oscillations 

 under the action of any forces which may or may not possess a 

 force-function, and is subject to resistances which vary as the 

 velocities of the parts resisted, the general equations of motion are 

 linear; and if ,v=Me mt , &c. their solution depends on a determi- 

 nantal equation 



/(m)= 



A, B, C. . . . 



a', b', a,... 



=0, 



the constituents being all of the form A=A 2 m 2 + A 1 m+A ; and if 

 the system has n degrees of freedom, /(m) is of the order 2n. " If 

 the roots of this equation are all unequal, the motion will be stable 

 if the real roots and the real parts of the imaginary roots are all 

 negative or zero, and unstable if any one is positive. If several 

 roots are equal, the motion will be stable if the real parts of those 

 roots are negative and not very small, and unstable if they are 

 negative and small, zero, or any positive quantity. But if, as often 

 happens in dynamical problems, the terms which contain i as a 

 factor are absent from the solution, the condition of stability is that 

 the real roots and real parts of the imaginary roots of the subsidiary 

 equation should be negative or zero." (P. 10.) 



When the system has two degrees of freedom the equation 

 f(ni)=Q is biquadratic; and this case has been worked out com- 

 pletely in the third edition of the Author's treatise on Rigid Dy- 

 namics (pp. 345, ....). In the present case about a third of the 

 Essay is devoted to a consideration of methods by which, without 



