HARMONIC ANALYSIS. 



j, a method by which the solution of any actual problem 



mar be obtained M the sum or resultant of a number of terms, each of which 

 ie "a solution of a particular case of the problem. The nature of these par- 

 ticular cases is defined by the condition that any one of them must be con- 

 jugal* to any other. 



The msthf^t^rfi 1 test of conjugacy is that the energy of the system 

 rising from two of the harmonics existing together is equal to the sum of 

 the energy arising from the two harmonics taken separately. In other words, 

 no put of the energy depends on the product of the amplitudes of two dif- 

 fcrant harmonic*. When two modes of motion of the same system are conju- 

 gate to fw4i other, the existence of one of them does not affect the other. 



The simplest case of harmonic analysis, that of which the treatment of 

 the vibrating string is an example, is completely investigated in what is known 

 M Fourier's Theorem. 



Fourier's theorem asserts that any periodic function of a single variable 

 period p, which does not become infinite at any phase, can be expanded in 

 the form of a series consisting of a constant term, together with a double 

 series of terms, one set involving cosines and the other sines of multiples of 

 the phase. 



Thus if j() is a periodic function of the variable f having a period p, 

 then it may be expanded as follows : 



The part of the theorem which is most frequently required, and which 

 also is the easiest to investigate, is the determination of the values of the 

 A, A, B,. These are 



P o 



t 2 

 '" 



