90 DYNAMICAL THEOKY OF SOUND 



Thus 



2 / 1 



f(x) = -j- I sin a sin x 4- ~- 2 sin 2a sin 2x 



+ i 2 sin3asin3#+...y ...(12) 



As a check on this result we may put a = JTT, x \ir\ this 

 gives 



which is known to be right. This example is of interest in 

 connection with the theory of the plucked string ( 26, 36). 

 Fig. 35 shews the graph of f(x) together with that of the 

 function represented by the first eight terms of the series on 

 the right hand of equation (12), in the case of a = f?r. The 

 fourth and eighth terms contribute nothing to the result in 

 this case, since they correspond to modes having a node at the 

 point plucked. 



Again, let /(#) = TT a? ...................... (14) 



2 f 77 2 



We find A s = \ (TT #)sins#c& = - ............. (15) 



TTJO s 



The theorem therefore asserts that 



TT - x = 2 (sin x + \ sin 2# -f sin 3# +...). . . .(16) 

 If we put x \ TT, we obtain 



which is Euler's formula for the quadrature of the circle. 

 The formula (16) also verifies obviously for #=TT; but if we 

 put x we see that there is some limitation to its validity. 

 The necessary modification is stated in 34. The series is 

 moreover much more slowly convergent than in the preceding 

 case; this is illustrated by Fig. 36, which shews the graph 

 of TT x together with that of the function represented by 

 the first eight terms of the series. For any value of x other 

 than we can obtain an approximation as close as we please, 

 provided we take a sufficient number of terms, but the smaller 

 the value of x the greater will be the number of terms required 

 to attain a prescribed standard. 



