46] EQUATION FOR PERIODS. 167 



are satisfied by 



A s = P smsft, 



where P is a constant, making use of the same trigonometric for- 

 mula as before. Accordingly let us substitute in the differential 

 equation 



ma d l y 



84) - y r _ t + ^ -^ + 2y r - y r+1 = 



the solution 



93) y r = Psinr-frcos^ s). 



Every term will contain the same cosine, so that dividing out we have 



- sin(r- 1)# + 2 l -~ - sinr# - sin(r + l)fr = O r 

 which is an identity if 



giving 



o a & /^ _ \ 



v 2 = (1 cos#), 

 ma v 



as before, 92). The complete solution is then 



s = n 



94) y r =^P S sin ^ cos (v f - ,), 



S=:l 



with the 2w arbitrary constants P g , cc 5 to be determined by the initial 

 displacements and velocities. 



Consider the case first in order of simplicity, n equals 2. Then 



->-\nr . * -I/IT 



v = 2 I/ sin = I/ ) 

 . * \ ma 6 r ma 



95) 



o - 



2 I/ - - sm = 

 ma 3 



Thus the frequency of the higher pitched vibration is in the ratio 

 of ~/3 : 1 = 1.732 to that of the lower, somewhat more than the 

 musical interval of a sixth. In this particular case it is easy to find 

 the normal coordinates. Writing 



96) 



<Pz = ^ (2/i - 2/2); 2/2 = ^i 



we obtain 



2 i =f(y; 2 +2/; 2 )=('p; 2 



97 ) <? 



S % 



