166 



V. OSCILLATIONS AND CYCLIC MOTIONS. 



Expanding the determinant in terms of its first minors we have 



This equation between three consecutive determinants of the same 

 form suggests a trigonometric relation, namely, making use of the 

 relation 



sin (a -j- &) -f sin (a 6) = 2 sin a cos Z>, 



with 1} = ft, a = n&, we have 



sin (n + 1) # -f- sin (n 1) # = 2 sin w# cos #. 

 Comparing this with the formula 89), 



we see that they are identical if we put 



C = 2 cos #, D n = fc.sin (n -f 1) #, 

 where c is independent of n. To find it put n = 1, 



sin 2# 



A 



2 cos 



Accordingly 

 90) 



Dn = 



sin (w -f- 1) fl- 



am # 



If this is to vanish we must have 



where Jc is any integer (not a multiple of n -f 1, to prevent sin # in 

 the denominator from vanishing). Introducing the values of # thus 

 found we obtain 



91) 

 from which 



92) 





 == 2 



\ 



\ 



Fig. 37. 



87) 



~ ~ 



= 2 cos # = 2 cos 



COS 



n+l 



Letting & = 1, 2, 3, . . . n, we obtain n different 

 frequencies proportional to the abscissae of 

 points dividing a quadrant into (n + 1) equal 

 parts, Fig. 37. Giving ~k other values not 

 multiples of (n + 1), we shall merely repeat 

 these frequencies. There are accordingly n 

 different frequencies for the vibrations. 



We may arrive at the same result by 

 noticing that the linear equations for the -4's, 



CA r - 



= 0, 



