Art. XIX. — On mi Expeditious Practical Method 

 of Harmonic Analysis} 



By THOMAS R. LYLE, M.A., 



Profeesor of Natural Philosoijhy in the University of Melbourne. 



(With Plates XXXI.-XXXIII.). 

 [Read 8th December, 1904]. 



1. Fourier has shown that if any function /{t) (=j say) of a 

 variable / be such that 



/(/)=/(/ + T)=/(/ + 2r) = etc., 

 where r is a constant, that is, if /(/) be periodic in /, of period t, 

 then/(/') can be expressed as the sum of a constant and a series 

 of terms called harmonics, each of the form 



apsin/>{(x}t - Op), 

 where fi has the values 1, 2, 3, 4, etc., 

 and (3} = ^ttJt. 

 The number/ is called the order of the harmonic, Op its ampli- 

 tude, and 6p its phase. 



If, in addition, /"(z") be such that 



then it is easy to see, by substituting t + TJ'2 for t, i.e., wZ + tt for 

 *at in 



y = a^-\-1ap^\np{oit—dp), 

 that in order for yt to be = —ytj^ri-i 

 Oq — 0, a^ = 0, ^4 = 0, etc. 

 Hence in this case the constant term vanishes and the har- 

 monics, of which /(/) is the sum, are all of odd order. When 

 such is the case /(i) is called an odd periodic function. This is 

 the type generally met with in alternating electric current in- 

 vestigations. 



1 Appendix to the paper : " Preliminary Account of a Wave Tracer and Analyzer." 

 Phil. Mag., Nov., 1903. 



