46 DYNAMICAL THEORY OF SOUND 



to. If, by the introduction of frictionless constraints which do 

 no work, the system be restricted to vibrate in a mode only 

 slightly different from one of these, the period will be altered 

 only by a small quantity of the second order. In other words 

 the periods of the several normal modes are " stationary " when 

 compared with those of slightly different constrained modes. 

 Suppose, for instance, that the normal mode in question is such 

 that in it the coordinate q l alone varies. We have, then, in (2), 

 a 12 = 0, c 12 = 0, and the natural frequency is determined by 

 n* = c u /a 11 . If the constraint be expressed by q 2 = \qi, the 

 condition that the constraining forces shall do no work, viz. 

 ftfc + Q2fc = 0, or ft + XQ 2 = 0, leads to 



(a 11 + X s toa)& + (c 11 + X s c B )g 1 = 0, ............ (4) 



and the speed (p) is accordingly given by 





When X is small, this differs from n 2 by a small quantity of 

 the second order. The proof, although limited to two degrees, 

 is easily generalized. Owing to our liberty of choice of the 

 coordinates, we can always arrange that q l shall be the only 

 coordinate which varies in the mode in question, and that 

 the constraint shall be expressed by a system of relations of 

 the type q 2 = \q l} q 3 = fj,q lf q 4 = vq lt .... 



For an obvious illustration we may have recourse again 

 to the particle on a smooth surface. If the constrained path 

 be a vertical section through the lowest point, the period is 

 %wJ(Rlg\ where R is the radius of curvature of the section, and 

 it is known that R is a maximum or minimum for the principal 

 sections. 



The equation (5) shews further that the constrained period is 

 (as in the particular case) intermediate between the two natural 

 periods ; this property can also be generalized. 



It follows that even when it is not easy to ascertain the 

 precise character of a particular normal mode, a close approxi- 

 mation to the frequency can often be obtained on the assumption 

 of an assumed type which we can judge on independent grounds 

 to be a fairly good representation of the true one. And in the 



