88 DYNAMICAL THEOKY OF SOUND 



The reasoning by which we have been led to this result is 

 of a physical rather than a mathematical nature, and we have 

 moreover not referred to the restrictions which physical con- 

 siderations alone would impose on the character of the arbitrary 

 function f(x). Leaving such points for the moment, and as- 

 suming the theorem provisionally, we proceed to the deter- 

 mination of the coefficients. If we multiply both sides of (3) 

 by sin sx, and integrate from x to x = ir y we get on the right 

 hand a series whose general term is 



A r I sin rx sin sx dx 

 Jo 



= %A r l {cos (r s) x cos (r 4- s) x} dx. ...(4) 

 J o 



When the integers r, s are unequal this vanishes, since each 

 cosine goes through its cycle of values, positive and negative, 

 once or oftener within the range of integration. But when 

 r = s, the first cosine is replaced by unity, and the result is 

 Hence 



2 f v 

 A g = -\ f(x)$msxdx ................ (5) 



The process may be illustrated by a few examples. Take, 

 first, the case of 



/(*) = (-*), ..................... (6) 



which is represented by an arc of a parabola. We find, after 

 a series of partial integrations, 



2 C w 4 



A 8 =-\ x (TT x) sin sx dx = -(1 COSSTT). ...(7) 



7T J o TrS"* 



This is equal to or 8/?rs 3 , according as s is even or odd. The 

 theorem therefore becomes 



x (TT x) - ( sin x + sin 3# + - sin 5x + . . . ) . -(8) 



7T \ O O / 



If we put x J TT in this we obtain the formula 



32"" 3 3 5 3 '"' ' 



which is known on other grounds to be correct. The equality 

 in (8) may also be tested graphically. It is found that the 

 discrepancy between the graph of x (TT x) and that of the 



