Okk — Stability or Instability of Motions of a Perfect Liquid. 35 



A concrete physical example may be given. Eouth discusses the following 

 problem* : — " A body has a point which is in one of the principal axes at the 

 centre of gravity G fixed in space. The body is in steady motion rotating 

 with angular velocity n about OG, which is vertical. Find the conditions that 

 the motion may be stable.." 



When the deviations are made to vary as e'P\ the resulting equations are 

 [{A - C) n' + Mgh + Bf]l + (A + B - C) inpr, = 

 -{A + B- C) inpl + {{B-C)n^ + Mgh + Ap'^U = 0, 

 ^, T], being the direction-cosines of the vertical referred to OA, OB, and h the 

 height of G above 0. Eouth investigates the condition when A = B, a case 

 which could not be made to suit the present requirements. We may, 

 however, simplify what follows by supposing C = A, when the equations 

 become 



[Mgh + Bf) 1 + B inpn = 0, -B inp ^+{{B - A)n^+ Mgh + Ap' U = 0. 

 Evidently what is required for the possibility of a solution of the type cited is 

 that, in the two fundamental oscillations, the two values of the quotient of ^ by 

 rj (or else of ^ by »j) should be nearly equal, and yet the two values of p not 

 nearly equal. This requirement is satisfied if the two values of Bp- are small 

 compared with Mgh, and yet not nearly equal. The equation determining p is 



{Bp- + Mgh)[A2f + Mgh - {A-B)n^} - B'ny = 0. 

 Evidently it is necessary for stability that Mgh - [A - B) n~ should be 

 positive ; and we will suppose A > B. Considering the equation 



{p)" + a) {p' + j3) - 7i?' = 0, ^ 



where a, (i., y are positive, we see that if, for example, 



£ being small, the two values of p-, namely, 



I {4£ + 3£'- ± v/(4£ + 2eJ - 4£^-}, 



are real, positive, very unequal, and small compared with a. Applying the 

 above conditions to the case in point, they are equivalent to 



Mgh- {A- B) 71' B_^/ 

 Mgh ■ A ~ ''' 



Bhv 



AMgh V 

 which lead, by elimination, to the following relation between A and B:— . 



{B' - {A-B)A{1 + 2ey} = e'AB. 

 This gives a value for B/A which is nearly equal to (^/S - 1) 2. 



* " Stability of a given State of Motion," p. 64. 



