HARMONIC ANALYSIS. 801 



This part of the theorem may be verified at once by multiplying both 

 sides of (1) by dg, by cos - d, or by sin - dg , and in each case inte- 

 grating from to p. 



The aeries is evidently single-valued for any given value of It cannot 

 therefore represent a function of which has more than one value, or which 

 becomes imaginary for any value of It is convergent, approaching to the 

 true value of <() for all values of f such that if varies infinitesimally 

 the function also varies infinitesimally. 



Sir W. Thomson, availing himself of the disk, globe, and cylinder inte- 

 grating machine invented by his brother, Professor James Thomson, has 

 constructed a machine by which eight of the integrals required for the ex- 

 pression of Fourier's series can be obtained simultaneously from the recorded 

 trace of any periodically variable quantity, such as the height of the tide, the 

 temperature of the pressure of the atmosphere, or the intensity of the different 

 components of terrestrial magnetism. If it were not on account of the waste 

 of time, instead of having a curve drawn by the action of the tide, and the 

 curve afterwards acted on by the machine, the time axis of the machine 

 itself might be driven by a clock, and the tide itself might work the second 

 variable of the machine, but this would involve the constant presence of an 

 expensive machine at every tidal station. 



It would not be devoid of interest, had we opportunity for it, to trace 

 the analogy between these mathematical and mechanical methods of harmonic 

 analysis and the dynamical processes which go on when a compound ray of 

 light is analysed into its simple vibrations by a prism, when a particular over- 

 tone is selected from a complex tone by a resonator, and when the enormously 

 complicated sound-wave of an orchestra, or even the discordant clamours of a 

 crowd, are interpreted into intelligible music or language by the attentive 

 listener, armed with the harp of three thousand strings, the resonance of 

 which, as it hangs in the gateway of his ear, discriminates the multifold 

 components of the waves of the aerial ocean. 



VOL. II. 



101 



