36 



DYNAMICAL THEORY OF SOUND 



with centre C and radius CP, in the plane ACB, about AB as 

 an axis; and the principal radii of curvature at the lowest point 

 are E l - CO, R z = EO, where E is the point of A B vertically 

 above 0. The corresponding directions of vibration are re- 

 spectively in and perpendicular to the plane ABO. 



Fig. 16. 



PC 



Fig. 15. 



Fig. 17. 



Another very simple case is that of two equal particles M 

 attached symmetrically at distances a from the ends of a tense 

 string, whose total length is, say, 2 (a + 6), so that 26 denotes 

 the length of the central portion. One obvious mode of 

 simple-harmonic vibration is that in which the deflections of the 

 two particles are always equal and of the same sign (Fig. 16). 

 If P be the tension of the string, the equation of motion of 

 either particle is then 



*5s >?. (8) 



and the speed is therefore 



(4) 



In another mode the two deflections are equal in magnitude 

 and opposite in sign, so that the middle point of the string is 

 stationary (Fig. 17). The circumstances are therefore exactly 

 the same as in 6, and the speed is 



'P a + b\ 



^r)' (5) 



greater, as we should expect, than 7^. If we denote the 



