58 DYNAMICAL THEORY OF SOUND 



If we put /ju-p* = Rcos a, kp = Rsina, (30) 



this becomes z = ^e i(pt ~ a \ (31) 



the real part of which is 



x = ^cos(pt-a) (32) 



This may be compared, for brevity, with the process of 12. 



21. Historical note. 



The theory of vibrations has a long and rather intricate 

 history, in which Pure Mathematics and Mechanics have 

 reacted on one another with great advantage to the progress 

 of both sciences. Various special problems of great interest 

 had been solved by the Bernoullis, Euler*, and other mathe- 

 maticians, but it is to Lagrange -|- that we owe the general 

 theory of the small oscillations of a system of finite freedom 

 treated by means of generalized coordinates. The work of 

 Lagrange was purposely somewhat abstract in formj; the 

 full dynamical interpretation was reserved for Thomson and 

 Tait (Natural Philosophy, 1867), to whom we also owe the 

 now current terminology of the subject. The theory has 

 been very greatly extended by Lord Rayleigh, and systematic- 

 ally applied to acoustics as well as other branches of physics, 

 in various writings, most of which (down to the year 1896) 

 are incorporated in his Theory of Sound . 



* Leonhard Eoler, born at Bale 1707, died at St Petersburg 1783. He wrote 

 extensively on most branches of mathematics and mechanics, and fixed to 

 a great extent the notations now in use. 



t Joseph Louis Lagrange, born at Turin 1736, died at Paris 1813, "the 

 greatest mathematician since the time of Newton." 



J "On ne trouvera point de Figures dans cet Ouvrage. Les m^thodes que 

 j'y expose ne demandent ni constructions, ni raisonnemens ge'ome'triques ou 

 me'chaniques, mais seulement des operations alge"briques, assujeties a une 

 marche reguliere et uniforme." (Preface to the Mecanique Analytique, 1788.) 



1st ed. London 1877, 2nd ed. London 1894 6. See also his Scientific 

 Papers, Cambridge 18991902. 



