326 TIDAL WAVES. [CHAP. VIII 



Substituting, we find 



^\A, + (a 2 X 2 + c 2 ) A 2 + 23X^3 + . . . = 0, 

 ^ S1 \A, + j3 32 \A 2 + (a 3 X 2 + c,) A 3 + . . . = 0, 



Eliminating the ratios A 1 : A z : A 3 : ..., we get the equation 



& 2 X, 



X, ... 



= (10), 



or, as we shall occasionally write it, for shortness, 



A(X) = ........................... (11). 



The determinant A (X) comes under the class called by Cay ley 

 'skew-determinants,' in virtue of the relation (6). If we re- 

 verse the sign of X, the rows and columns are simply interchanged, 

 and the value of the determinant therefore unaltered. Hence, 

 when expanded, the equation (10) will involve only even powers of 

 X, and the roots will be in pairs of the form 



In order that the configuration of relative equilibrium should 

 be stable it is essential that the values of p should all be zero, 

 for otherwise terms of the forms e p * cos at and e pt sin.at would 

 present themselves in the realized expression for any coordinate 

 q s . This would indicate the possibility of an oscillation of 

 continually increasing amplitude. 



In the theory of absolute equilibrium, sketched in Art. 165, 

 the necessary and sufficient condition of stability is simply that 

 the potential energy must be a minimum in the configuration of 

 equilibrium. In the present case the conditions are more com- 

 plicated*, but we may readily shew that if the expression for 

 V TQ be essentially positive, in other words if the coefficients 

 Cj, C 2 , ... in (4) be all positive, the equilibrium will be stable. 

 This follows at once from the equation (7), which gives, in the 

 case of free motion, 



-T ) = const ......... .......... (12), 



* They have been investigated by Routh, On the Stability of a Given State of 

 Motion ; see also his Advanced Rigid Dynamics (4th ed,), London, 1884. 



