Mr Harrison, Notes on the Theory of Vibrations 87 



The determinantal equation for the periods is 

 A2 (1 + ^j2) _ ;^^2^ Aii/^A2, /^i/^A2, .. 



flltl2^^, A2 (1 + IL^^) - Ag^, iUg.UgA^, ., 



= 0. 



;i) 



Let Aj2, Ag^, ... be arranged in ascending order of magnitude. 

 If A^ = 0, the left-hand side of (1) is (— 1)" as regards sign. If 

 A^ = Aj^, the left-hand side of (1) is equal to 



fh' 



V 





 











and this is (-- 1)"-^ as regards sign. 



Hence all the roots in A^ are decreased and they are separated 



The validity of this proof depends on (1) the non-equahty of 

 any of the values of A^^, Xo^, ..., (2) the non-evanescence of any of 

 the constants /Aj, /Xg, .... In case of (J) one period at least of the 

 modified system is equal to a period of the original, but the theorem 

 may be held to cover this case. 



In case of (2) the theorem does not remain true. Suppose 

 the ju.'s are all zero except /x^, /Xg, f^t^ •••• Then only the periods 

 corresponding to q,., qg, q^, ... are changed. The periods belonging 

 to these coordinates will be increased and their new values will be 

 separated by their old values. But these new periods bear no 

 relation to the periods belonging to the remaining coordinates and 

 can occupy any position in regard to them except as specified 

 above. Hence the theorem does not seem to indicate where the 

 modified periods must lie in regard to the complete system of 

 periods of the undisturbed system. 



An example is afforded by the modification introduced into 

 the periods of a stretched string by a load attached at a point 

 dividing the string into two lengths which are commensurable. 

 Rayleigh's argument (vide Theory of Sound, Vol. i, p. 122), which 

 serves to maintain the validity of the theorem in this case, is 

 acceptable owing to the strictly defined relations which exist be- 

 tween the periods in both states. But in an ordinary dynamical 

 problem the theorem must be held to break down in the excep- 

 tional cases under consideration since it fails completely to indicate 

 the position of the modified periods in relation to the original 

 periods. 



